In DDO, just like in tabletop D&D, classes represents the identity and nature of the characters that you a playing. A characters capabilities, strengths, and weaknesses are defined by its class. For most races within the game, you must choose a class during the creation or reincarnation process. For Iconics, the first class is automatically chosen during character creation.
As a character levels up, you can choose to have more than one class. At any single time during the life of a character, you can at most 3 classes (called multi-classing).
There are currently 14 classes to choose from in DDO.
For more information about a specific class, click on the links above.