Standard Attack
The most common action to take in combat is the Attack action, whether you are swinging a sword, firing an arrow from a bow, or brawling with your fists.
With this action, you make one melee or ranged attack. See the "Making an Attack" section for the rules that govern attacks.
Certain features, such as the Extra Attack feature of the fighter, allow you to make more than one attack with this action.
Disarm
A creature can use a weapon attack or unarmed strike to knock a weapon or another item from a target's grasp. The attacker makes an attack roll contested by the target's Strength (Athletics) check or Dexterity (Acrobatics) check. If the attacker wins the contest, the attack causes no damage or other ill effect, but the defender drops the item.
The attacker has disadvantage on its attack roll if the target is holding the item with two or more hands. The target has advantage on its ability check if it is larger than the attacking creature, or disadvantage if it is smaller.
Grapple
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them. The target of your grapple must be no more than one size larger than you, and it must be within your reach.
Using at least one free hand, you try to seize the target by making a grapple check, a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you subject the target to the grappled condition (see the appendix). The condition specifies the things that end it, and you can release the target whenever you like (no action required).
When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
Shove
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target of your shove must be no more than one size larger than you, and it must be within your reach. You make a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you either knock the target prone or push it 5 feet away from you.
Shove Aside
With this option, a creature uses the special shove attack from the Player's Handbook to force a target to the side, rather than away. The attacker has disadvantage on its Strength (Athletics) check when it does so. If that check is successful, the attacker moves the target 5 feet to a different space within its reach.
Sources: PHB, page 192 & DMG, page 271
Climb onto a Bigger Creature
If one creature wants to jump onto another creature, it can do so by grappling. A small or Medium creature has little chance of making a successful grapple against a Huge or Gargantuan creature, however, unless magic has granted the grappler supernatural might.
As an alternative, a suitably large opponent can be treated as terrain for the purpose of jumping onto its back or clinging to a limb. After making any ability checks necessary to get into position and onto the larger creature, the smaller creature uses its action to make a Strength (Athletics) or Dexterity (Acrobatics) check contested by the target's Dexterity (Acrobatics) check. If it wins the contest, the smaller creature successfully moves into the target creature's space, the smaller creature moves with the target and has advantage on attack rolls against it.
The smaller creature can move around within the larger creature's space, treating the space as difficult terrain. The larger creature's ability to attack the smaller creature depends on the smaller creature's location, and is left to your discretion. The larger creature can dislodge the smaller creature as an action—knocking it off, scraping it against a wall, or grabbing and throwing it—by making a Strength (Athletics) check contested by the smaller creature's Strength (Athletics) or Dexterity (Acrobatics) check. The smaller creature chooses which ability to use.
This action is an optional addition to the game, from the optional/variant rule Action Options.
Source: DMG, page 271
