When you reach a certain amount of Experience Points (gained from proving your competence at a variety of adventuring-related tasks such as defeating foes, completing quests and successfully passing some skill checks) you'll level up. When you level up, you can increase your Class Level in any class you meet the qualifications for. Your cumulative Class Levels - the total number of times you've leveled up - is your Character Level. Your Character Level, then, is a rough approximation of your character's general level of experience, competence and power.
When your Character Level increases, you'll gain more Hit Dice and Hit Points, and depending on your Character Level and Class Level, you may gain Feats, better Saving Throws, a higher Base Attack Bonus and new class abilities, as well.
Feat gains by Character Level are static - you'll gain a feat at 1st level, and at every odd level thereafter (3rd, 5th, 7th, etc.), but you may also gain extra feats from your Class Levels. For example a Fighter gains a bonus feat at 1st level, 2nd level, and every even level (4th, 6th, 10th, etc) thereafter. These bonus feats from a Fighter's Class Levels are in addition to any gained from a character's overall Character Level.
Most other benefits of increasing your level are highly variable by your Class Level, and are usually directly additive. You'll always gain Hit Points when you level up, but your Class Level will determine your Hit Dice, and your Saving Throw and Base Attack Bonuses are determined solely by combining the bonuses gained by your Class Levels. For example, a 5th level Fighter would have a +5 Base Attack Bonus and +4 Fortitude, +1 Reflex and +1 Will and a 5th level Wizard would have a +2 Base Attack Bonus, +1 Fortitude, +1 Reflex and +4 Will. A 10th level (Character Level) who was a 5th level Fighter/5th level Wizard would thus have a +7 Base Attack Bonus, +5 Fortitude, +2 Reflex and +5 Will, not to mention ten Hit Dice (5d10 + 5d6).