A character's class affects a character's skills and abilities directly. For instance, a fighter is likely to have large amounts of hit points and possess great skill at attacking an opponent directly in physical combat, while a wizard would be physically frail yet have a selection of powerful magic spells with which to aid the party.

As a character gains experience points, they are likely to increase the "level" of their class. Each increase grants the bonuses of the next level, strengthening the character. Throughout the editions of Dungeons & Dragons, an increase in level has generally brought about increased hit points, more skills / proficiencies, a bonus to the accuracy of physical strikes, more magical spells for spellcasters, and better "saving throw" bonuses at resisting hostile magical effects. In addition, each level grants special abilities specific to the class; for example, a Paladin gains the ability to "Turn" (Repel) undead at a certain level.

In first and second editions, changing a character's class was difficult. Only humans could do it, and they had to meet some rather steep requirements to do so. However, non-humans could "multi-class" where they effectively learned two (or rarely even three) classes at the same time at the cost of a slower character level progression. 3rd edition allows characters to mix classes more easily; as each new level is attained, a player can choose to add a level in a character's existing class, or add the first level in a new class. There are, however, penalties to the rate of experience point gained if classes are added haphazardly.

In fourth edition, multiclassing is handled through feats: for each class, there is a single class-specific multiclass feat, available to members of any other class who meet the attribute prerequisites and have not already taken another class-specific multiclass feat. A character who takes a class-specific multiclass feat gains certain class features from that class, is considered a member of that class for the purposes of meeting pre-requisites for feats and paragon paths. The class-specific multiclass feats are also pre-requisites for the power-swap feats, each of which allows to swap out a daily, encounter or utility power from their first class for one from their second class. At 11th level, character can forego a paragon path in order to further specialize in their second class, providing unspecified benefits.

Taken from Wikipedia