Each skill represents a different field of expertise. Some skills have different sub-types (e.g. the craft skill is divided into craft armor and craft alchemy -- among others), while others are used by a single task (e.g. attempting to stealth requires a good proficiency in both hide and move silently). Certain skills provide a synergy which expertise in one field provides better understanding of a different skill (e.g. being able to craft a trap gives you some insider knowledge on how to dismantle a trap).
An amount of skill points are gained at each level with the amount gained depending on which class is being levelled. These points can be used to increase the rank of any skill. However, some skills are more easily understood by some classes. While class skills (those a class can easily understand. e.g. a wizard understands skills related to spell casting) provide 1 skill rank for each skill point invested, cross-class skills are more expensive to represent the difficulty of learning something less associated with a class' basic training (e.g. a wizard has a harder time understanding stealth abilities than a sneaky rogue would).
The check works just like an attack roll or a saving throw -— the higher the roll, the better. Either you're trying to match or exceed a certain difficulty class (DC), or you’re trying to beat another character's check result, i.e., an opposed skill check.