Ability Checks are tests of your characters skills and abilities, based upon your Attributes. An Ability Check may be queried by your Game Master for any action, usually where the result of the action taken is uncertain.
The GM determines which Attribute is used in an Ability Check depending on the task at hand. Ability Checks have a chance of failure, and the difficulty of the task is represented by the Difficulty Class, or DC, of the check.
To make an Ability Check, roll a d20 dice, and add the Attribute Bonus of the queried Attribute to the result of the roll. If the total equals or exceeds the DC, the ability check is a success, and the creature making the check performs the task. If the total is lower than the DC, it is a failure, and the GM may determine setbacks, or other complications and downsides.
Skills represent Attributes being used for specific purposes and applications. Your skills gain bonuses from several sources, such as Skill Points, and class features.
It should be noted that ability checks should be used as a way to facilitate a character doing something that has an uncertain result or effect, or potential degrees of success due to variables, and not a default option when a character decides to do something.