Skills increase your chance of success on specific kinds of actions and tasks. Some give other passive bonuses.
Training Levels
Each Skill starts at the Novice level and can be increased to Trained and Specialized, granting greater bonuses at each level. Descriptors often leave some Skills at the Inability level, meaning they inflict penalties. When upgraded, these return to the Novice level.
The bonuses are generally as follows:
Inability: -15% to Effort success
Novice: No bonuses
Trained: +15% to Effort success
Specialized: +30% to Effort success

Weapon Skills allow you to use weapons more effectively in battle, increasing your chances to hit your targets (+15% per level). Only Glaives receive training in weapon skills, but appropriate descriptors can confer a degree of knowledge on you as well.
Light Weapons
Medium Weapons
Heavy Weapons
Melee Weapons
Ranged Weapons

Exploration Skills provide benefits to any tasks and challenges you may come across, increasing your odds of success while using them. Some Exploration Skills may also have additional benefits, such as granting increased Health, or increasing the number of Cyphers you can safely carry.
Endurance
Quick Fingers
Running
Smashing
Anamnesis
Concentration
Cypher Use
Healing
Perception
Stealth
Deception
Intimidation
Persuasion
Initiative
Tidal Affinity

Lore Skills focus on the theoretical applications of knowledge.
Lore: Natural
Lore: Mystical
Lore: Machinery