Level

Your level can refer to many things, either your class level your character level or also spell level or even dungeon level.

Class level
Your class level is a measure of how far you have advanced in that particluar class. For example of you were a 10th level Barbarian and a 6th level fighter then you would have 10 class levels in barbarian and 6 class levels in fighter.

Character Level
Character level refers to the sum of your class levels. In the above example you would have a character level of 16. This is important in determining feats and also ability score increases as they are based off of character levels.

Caster level
Your class level in the class you are casting a spell with. See the specific caster level page for details.

Experience Points
To advance your level, you need to gain experience points from doing quests. Progression table is here.

Spell Level
Spells are also broken into levels based on general strength. Spell levels range from 1 (the weakest) to 9 (the strongest). Only dedicated casting classes have access to nine full levels of spells; the strongest Bard spells are only level 6, and Paladin and Ranger spells stop at level 4. Spell levels do not match character or caster levels one-for-one. It can take two, three, or even four caster levels to gain access to a character's next level of spells.

Other uses of level
Dungeons are usually grouped into levels separated by staircases. The deeper you are in the dungeon the tougher the monsters are going to be.