Classes in EverQuest Next will work a bit differently compared to what is prelevant in MMOs. At the beginning of the game, players will become one of eight traditional classes and as they get out into the world, they can "collect" over 40 other classes, allowing them to mix and match to make their characters fully customizable, through the mechanic of multi-classing. The SOE Live reveal only discussed these, but we got to see the Warrior and the Wizard. Other classes we've heard of so far: Monk, Paladin, Blademaster, Ranger, Tempest and Rogue.
What classes a player has equipped determines what armor and weapons may be used. Classes are exchangeable out of combat. Players will have the ability to name their custom classes. There is currently no artificial limitation toward how many classes a player may collect. EverQuest Next also aims to have less of a "holy trinity" with no dedicated tanks, healers, threat meters and the like.
For more information, please see the EQNext Fans write-up, complete with videos!