The game rolls an Ability Check to determine whether a creature will succeed or fail in various scenarios. The creature rolls a D20, and adds its Ability Score Modifier and possibly a Proficiency Bonus. To succeed, the result must reach or exceed a Difficulty Class, which depends on what the creature is trying to accomplish.
Ability Checks can be based purely on an Ability Score, in which case they can only benefit from the Ability Score Modifier. More commonly, they will be a Skill Check, where the Proficiency Bonus can be added as well if the creature is Proficient in the given Skill.
For example, when a trap is nearby, the game rolls a Perception Check (or in short, "rolls Perception"). Perception is related to Wisdom, so the creature's Wisdom Ability Score Modifier is applied. If the creature is Proficient in Perception, then the creature's Proficiency Bonus is also added.
For example, all Elves have an innate feature called Keen Senses which grants them Proficiency in Perception. Wood Elves have an additional feature called Mask of the Wild, which gives them Proficiency in Stealth. Rangers can gain Proficiency in a number of skills based on specific class features they choose at level 1: Bounty Hunter grants Proficiency in Investigation, Keeper of the Veil and Mage Breaker grant Proficiency in Arcana, and so on.
The Skilled Feat, if chosen when leveling up, allows a creature to gain Proficiency in any three Skills.
Following is the list of Skills that a creature can be Proficient in, sorted by the Ability Score which the skill is related to:
A special type of Ability Check is called the Contest, in which two creatures both roll an Ability Check or Skill Check to contest each other, and one wins over the other. The creatures don't necessarily roll the same type of check.
An example of this is the Shove action, where the creature attempting the Shove must roll Athletics, and the defending creature is allowed to roll either Athletics or Acrobatics to contest the Shove. If the attacker's roll is higher than the defender's, the Shove succeeds, otherwise it fails.
For reference, here are once again the Ability Score Modifier and Proficiency Bonus tables:
|Ability Score||Modifier Value||Ability Score||Modifier Value|
|1 to 4||+ 2|
|5 to 8||+ 3|
|9 to 12||+ 4|