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Some background filler:
Classes in earlier versions of D&D used to have some fairly hefty requirements to join them, even for a single classed PC, Str 9 for a Fighter, Int 9 for a Wizard, multiple minimum stats for Paladins and Rangers, etc.
With Pathfinder, most of those restrictions went away. Some restrictions remain, like Paladins needing to be LG, Monks, mostly, needing to be Lawful, Barbarians needing to be non-Lawful, etc.
For your PC, you have, as mentioned, Class Levels and Character Level.
Class Levels are the levels your PC has in a specific class, like Cleric or Rogue.
Character Level is your PC's total class levels added together.
Cleric 2/Rogue 1 would be:
Cleric level 2, Rogue level 1, Character level 3.
As a Cleric, he casts spells as a 2nd level Cleric, including spell slots, etc.
As a Rogue, he would only gain the benefits of the first level of Rogue.
Character level 3 means that he gets the new feat.
If he mixes two spellcasting classes together, each class gets its own spell slots, spells known (if applicable), and caster level.
Wizard 3, Cleric 3 would have two sets of spells, as a third level Wizard and a third level Cleric.
Prestige classes (aka PrC), in general, have requirements built-in that give them a de facto minimum level requirement to join, whether it is BAB +5, 5 ranks in skill X, Y, and Z, or being able to cast 3rd level spell from a single class, or second level spells from two different classes. Remember that you cannot use anything from the Prestige Class itself to qualify for the Prestige Class, so no taking a level of Arcane Archer to get the skill ranks needed to qualify to become an Arcane Archer.
Example:
Needed to join: 5 ranks in Knowledge (religion)
As you get skill ranks from your class, you have to have 5 levels in other classes in order to qualify, so you cannot join this PrC at 5th level, as you do not yet have 5 ranks in any skill.