Known Skills and Tools
All skills and tools have a typical ability score associated with them. This skills and tools list below all have that ability score in parentheses. This is the normal case, but almost any skill can have a different skill associated from time to time. Climbing a tree is always an athletics check, but maybe it is more about your fingers finding the correct handholds making a wisdom or dexterity based instead of strength based. Or maybe a thief is testing your rogue’s knowledge of hiding in plain sight. That might make your stealth roll based on intelligence and not dexterity. Standard Skills
As described in the Player's Handbook, a skill proficiency represents a character's focus on one aspect of an ability. Among all the things a character's Dexterity score describes, the character might be particularly skilled at sneaking around, reflected in proficiency in the Stealth skill. When that skill is used for an ability check, it is usually used with Dexterity.
Under certain circumstances, you can decide a character's proficiency in a skill can be applied to a different ability check. For example, you might decide that a character forced to swim from an island to the mainland must succeed on a constitution check (as opposed to a Strength check) because of the distance involved. The character is proficient in the Athletics skill, which covers swimming, so you allow the character's proficiency bonus to apply to this ability check. In effect, you're asking for a Constitution (Athletics) check, instead of a Strength (Athletics) check.
Often, players ask whether they can apply a skill proficiency to an ability check. If a player can provide a good justification for why a character's training and aptitude in a skill should apply to the check, go ahead and allow it, rewarding the player's creative thinking.