There is not much information about quests and quest mechanics for Diablo 3.
Of course there will be quests, which will be essential for the main plot.

The quantity of quests will be much larger than Diablo 1 or Diablo 2, because blizzard tries to implement more roleplaying features without losing the “hack ‘n slay” focus.

If it will be possible to ignore some of these quests, and still be able to advance in the game, or not, like the Quest “Bloodraven” in Diablo 2 is yet unknown.

Various randomized quests will ensure the possibility to play the game over and over again, as we are accustomed from all games from the Diablo series.
Scripted adventures will be generated random all over the world, e.g. like a caravan, that needs escort, or monstercamps, which are always led by a bossmob.

A new quest related feature are the mentioned character specific quests. How they will work, or in which way they are focused on the special skills of each character, is also not yet revealed.

It could be possible that some quests can only be solved by certain characters.

For example a barbarian could help a NPC, who is trapped over a gaping canyon.
Other classes will not be able to solve this quest, because they don’t have the needed abilities, but the barbarian could leap over the canyon to rescue the NPC.