Alignment and Its Impact

Mac: Guys, why aren't the brakes working?!
Charlie: Because I cut the brakes! Wild card, bitches! YEEEEEEEEHHHAAAAWWWW!!! [leaps from van]
-Always Sunny In Philadelphia

Alignment is a bit of a strange topic; different versions of D&D have treated it with varying degrees of importance, criteria, and associated mechanics. To avoid making this too confusing, I'm going to stick to two areas; how 5th Edition specifically defines and refers to alignment, along with some pretty common scenarios, mechanics, and concerns that your DM may have in relation to Alignment.
To get this out of the way; previous editions of D&D put a very strong emphasis on alignment when compared to the current 5th edition. When looking up information for yourself on alignment mechanics, make sure that you're looking at the 5th edition ruleset. Your DMs both referenced these previous rulesets when we were discussing this topic post Session 9, and I think it may have made things more confusing.

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Alignment in 5th Edition Dungeons and Dragons

Character Alignments get mentioned over and over throughout the Player's Handbook and other Reference Materials, however they are only DEFINED in a few places.

Alignment Descriptions within the Player's Handbook

This information can be found on page 122 of the Player's Handbook (PHB), or any online resource of the 5th Edition System Reference Document (SRD).
Lawful good (LG) creatures can be counted on to do the right thing as expected by society. Gold dragons, paladins, and most dwarves are lawful good.
Neutral good (NG) folk do the best they can to help others according to their needs. Many celestials, some cloud giants, and most gnomes are neutral good.
Chaotic good (CG) creatures act as their conscience directs, with little regard for what others expect. Copper dragons, many elves, and unicorns are chaotic good.
Lawful neutral (LN) individuals act in accordance with law, tradition, or personal codes. Many monks and some wizards are lawful neutral.
Neutral (N) is the alignment of those who prefer to steer clear of moral questions and don't take sides, doing what seems best at the time. Lizardfolk, most druids, and many humans are neutral.
Chaotic neutral (CN) creatures follow their whims, holding their personal freedom above all else. Many barbarians and rogues, and some bards, are chaotic neutral.
Lawful evil (LE) creatures methodically take what they want, within the limits of a code of tradition, loyalty, or order. Devils, blue dragons, and hobgoblins are lawful evil.
Neutral evil (NE) is the alignment of those who do whatever they can get away with, without compassion or qualms. Many drow, some cloud giants, and goblins are neutral evil.
Chaotic evil (CE) creatures act with arbitrary violence, spurred by their greed, hatred, or bloodlust. Demons, red dragons, and orcs are chaotic evil.

Alignment in the Multiverse

For many thinking creatures, alignment is a moral choice. Humans, dwarves, elves, and other humanoid races can choose whether to follow the paths of good or evil, law or chaos. According to myth, the good-aligned gods who created these races gave them free will to choose their moral paths, knowing that good without free will is slavery. The evil deities who created other races, though, made those races to serve them. Those races have strong inborn tendencies that match the nature of their gods. Most orcs share the violent, savage nature of the orc gods, and are thus inclined toward evil. Even if an orc chooses a good alignment, it struggles against its innate tendencies for its entire life. (Even half-orcs feel the lingering pull of the orc god's influence.) Alignment is an essential part of the nature of celestials and fiends. A devil does not choose to be lawful evil, and it doesn't tend toward lawful evil, but rather it is lawful evil in its essence. If it somehow ceased to be lawful evil, it would cease to be a devil.
Most creatures that lack the capacity for rational thought do not have alignments - they are unaligned. Such a creature is incapable of making a moral or ethical choice and acts according to its bestial nature. Sharks are savage predators, for example, but they are not evil; they have no alignment.

Soft References and Inferences

The above definitions of the alignments are, by most measures, lacking. Beyond that, constant mentions of alignment in other sections of the PHB imply a greater importance. I am including all of these references as a means of further fleshing out what each alignment is supposed to represent, as stated by the hard ruleset.
In the Player Race Descriptions:
Dragonborn: Dragonborn tend to extremes, making a conscious choice for one side or the other in the cosmic war between good and evil. Most dragonborn are good, but those who side with evil can be terrible villains.
Dwarves: Most dwarves are lawful, believing firmly in the benefits of a well-­‐‑ordered society. They tend toward good as well, with a strong sense of fair play and a belief that everyone deserves to share in the benefits of a just order.
Elves: Elves love freedom, variety, and self-expression, so they lean strongly toward the gentler aspects of chaos. They value and protect others’ freedom as well as their own, and they are more often good than not. The drow are an exception; their exile has made them vicious and dangerous. Drow are more often evil than not.
Gnomes: Gnomes are most often good. Those who tend toward law are sages, engineers, researchers, scholars, investigators, or inventors. Those who tend toward chaos are minstrels, tricksters, wanderers, or fanciful jewelers. Gnomes are good-hearted, and even the tricksters among them are more playful than vicious.
Half-Elf: Half-elves share the chaotic bent of their elven heritage. They value both personal freedom and creative expression, demonstrating neither love of leaders nor desire for followers. They chafe at rules, resent others’ demands, and sometimes prove unreliable, or at least unpredictable.
Half-orc: Half-orcs inherit a tendency toward chaos from their orc parents and are not strongly inclined toward good. Half-orcs raised among orcs and willing to live out their lives among them are usually evil.
Halfling: Most halflings are lawful good. As a rule, they are good-hearted and kind, hate to see others in pain, and have no tolerance for oppression. They are also very orderly and traditional, leaning heavily on the support of their community and the comfort of their old ways.
Human: Humans tend toward no particular alignment. The best and the worst are found among them.
Tiefling: Tieflings might not have an innate tendency toward evil, but many of them end up there. Evil or not, an independent nature inclines many tieflings toward a chaotic alignment.

In the Player Class Descriptions:
Only a few classes make reference to Alignments.
Monk: Page 77 of the PHB makes reference that Monks are almost always Lawful in their alignment.
Paladin: Page 83 of the PHB makes references to how Paladins are rarely evil, and the importance of how their alignment and oath relate to each other.
In the Character Backgrounds:
Interestingly, perhaps the most definition we get for alignments come through the various examples of Character Ideal Tables found in the PHB Section on Backgrounds, starting on page 126. There are too many to list here. Note the Alignment Characteristic in parenthesis at the end of each ideal in the various background tables to gain further perspective on the sorts of acts and opinions that are associated with each alignment.
Gods and Planes:
Here is perhaps the most overt connection to in-game mechanics suggested in the PHB (Appendix B, p293 onward), alluded to in the "Alignment in the Multiverse" Section above, and which hints at a lot of what I'll be describing in the next section. Gods in D&D are a reality, and gods are traditionally defined by their ideals, domains, and alignments. Interactions with a god or their church will be heavily defined by how closely your actions and ideals align with theirs.
Along with the gods, outer planes are defined in the PHB (Appendix C, p 302) as being associated with alignments as well. The outer planes are where the gods reside, and the alignment of a plane can have adverse effects on its inhabitants, although the natures of such are not necessarily known by mortals.

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Alignment from the DM's Perspective

Now knowing how broadly and vaguely the official material talks about alignment, here's what you can expect from your DMs about if (let alone why) alignment could be important.

A character dictates their alignment, not the other way around.

I think this, conceptually, got lost in the weeds a bit during our discussion on the topic, and a big part of why I suspect has to do with 3rd Edition. In 3rd Edition D&D (along with others, but 3/3.5 is the big one I'm familiar with), your alignment, and the alignment of creatures/people you encounter are an aspect of your character that has mechanical implications. Certain spells worked against evil, or good. An entire class (paladin) mandated that the player be consistently Lawful Good, or lose their paladin powers. Your alignment could define your character. But this is not the case in 5th Edition. There are no spells, abilities, classes, or races, that MANDATE an alignment distinction. Note in the above sections for races, all the weasel words, like "most" or "tend to". This is because your character is first and foremost, whoever you deem them to be, and their alignment stems from that.

If it's not something that defines my character, why would I need to care about it?

Alignment is, for the player's purposes, more a shortcut tool to help them remain consistent in their roleplay. This is more embodied in 5th by the ideas of background flaws and ideals that can be chosen during character creation, or adopted over time as your character grows. Remembering a few key character traits that you decided your character has, allows you to more fluidly respond to new situations without having to deconstruct your character's backstory and motivations every time.
But then, enter your DM. As hinted at in some of the above sections, there are some hard rulesets within the game regarding people and places, and how they interact/impact people of certain alignments. This means that, even if you haven't decided that your character exists as a certain alignment, your DM has to, for mechanical purposes, assign them one. And so if, for whatever reason, you do or don't want your character to be perceived as a specific alignment, you need to coordinate with your DM to make sure you're both on the same page as to how that can be maintained. That said....

Good and Evil are just words. Loaded with implication, but just words.

If you read through the various ideals associated with good and evil, it becomes clear that certain character traits are associated with each. I know I (Zack) used selfless/selfish as an example of just one trait and how it spreads from good to evil, but there are many such traits, all considered together. But does being greedy really make them evil? Is a character evil for killing elves if they believe that, for example, the existence of elves will lead to the end of the world, and so elves must be destroyed to save everyone else? In both of those situations, assuming the character consistently demonstrates those traits, then yes, they would be. But that doesn't mean the character is necessarily doing something wrong.
Good and Evil are rigidly defined, but "right" and "wrong" in D&D is entirely dependent on who you ask. Some of the racial descriptions above hint at this as well. A thriving, successful society can function by espousing ideals that the handbook considers "evil", and there are innumerable gods that justify and bless just about every mindset imaginable. Whereas a character with an evil alignment may find themselves isolated or at odds with the law in a lawful good dwarven kingdom, a character with a good alignment will find themselves equally in trouble in say, a lawful evil vampire's domain. The god of storms looks fondly down upon the barbarian raging against the constraints of society. The goddess of murder takes every grisly slaughter as a personal gift. The god of knowledge really doesn't give a shit how you carry yourself, so long as you carry yourself in the pursuit of knowledge. So don't get too hung up on whether the dm calls your character "good" or "chaotic" or "evil," because it's all relative.

It's all relative, but some relations matter more

Up until this point, I've referred largely to alignment as a relationship to the gods. But a character's relation to the gods isn't all that important in the grand scheme. Conflict with gods is a lot of fun, and a staple of D&D. However, there are people far more important than the gods in D&D; the other players.
D&D is cooperative storytelling. Cooperation between the players and the DM, and between the players themselves. A group of like-minded adventurers can change the world, as the players are roleplaying towards a goal that their characters collectively wish to achieve. When a character alignment sits opposite the rest of the party, things can get tricky. Characters that inherently disagree with each other about the most basic of things can lead to regular infighting and frustration. Obviously it makes a compelling plot point, and can lead to some really unique character interactions. Short term instances of this make for good obstacles for the party to overcome. But if, say, the long term intended goal for a character is (pulling from before) the extinction of the elves, and they are playing with a group of traditionally "good" players (or some of them are elves), someone's going to have to budge. A cleric of life and charity is just setting themselves up to fail if they're a part of a group determined to overthrow Asmodeus and rule the nine hells. Yes, both of these can work, but the player has to recognize that, through character and story development, their character is going to have to change, and find a way to justify their continued association with the party, and vise versa. The elf-genocidest from before has a change of heart, realizing that their understanding of fate must be wrong, after being saved time and time again by their elven companion. The cleric determines that their mindset was naive of how the world works, and takes to the iron-fisted nature of those that thrive in the nine hells.
There are even ways to design a character that functions perfectly well within a group of opposing alignment, should the player be clever in designing their goals. An evil character, for example, calculating and patient, with a hyperspecific goal that does not negatively impact the good party, could happily go along with the party, gaining allies and favor, to eventually call those favors in for their personal goal. A good character in an evil party might be able to rationalize that whatever the evil goals are, are less evil than the status quo, and as such are willing to work with people willing to work with them.

Cooperation is key

If the player really had their heart set on their character's goals and motivations, and it just isn't going to fit with the rest of the party? Just be prepared for the inevitable showdown with your party members, likely resulting in your character either leaving the party, or death. And that's at best; campaigns have ended outright because of things like this. Be prepared and ready to create a new character when that happens, one that presumably jives a bit better with the party. None of this in itself is a problem, so long is everyone is on the same page. A character being angry is good story, a player being angry means a breakdown in communication. I (Zack) actually brought this topic up to the more veteran DMs that run the 3.5 game I play, and two of them said outright that they won't even allow evil player characters in their games, because of the inevitable infighting bleeding from the characters to the players, ruining the campaign. I don't think the problem is THAT severe, but that's the lengths some DMs go to avoid this potential problem.
A character unswayingly at odds with the rest of the party can lead to some interesting long term plot, so long as the player and DM are in agreement. The character forced out of the group could develop into a wonderful enemy that the party has to deal with in the future, and with personal ties to the party makes for juicy storytelling! The wizard bent on the destruction of the elves! The crusading cleric trying to put a stop to a threat they perceive as even more dangerous than Asmodeus! But this should only happen if everyone involved is okay with it.

TL:DR So what the hell am I supposed to take from all this?

At the end of the day, play your character as you want to, and only care/think about alignment if it's something you or your character think are important. If you have goals for the character that you worry may not gel with the party or campaign, talk to the other players and DM about it! So long as your character is getting along with the other player characters, alignment is more your DM's problem than yours. If you think that your character's story would be more interesting at odds with the rest of the party, make sure the DM knows that's the route you would like to go, so that they can set up interesting plot points and encounters to further emphasize this growing rift in an interesting and fulfilling way.