| Elementalists have the elemental spells (Fire Earth Water Air); Mesmers are more like illusionists, with spells that interrupt and steal energy (mana) from enemies.
There are 6 core classes (Elementalist, Mesmer, Monk, Necromancer, Ranger, Warrior) which (will) appear in every GW campaign, and Assassins and Ritualists which only appear in Factions.
Once you get far enough into the game, you gain the ability to change your secondary class whenever you want to. People like to switch secondarys around so that they can 'unlock' more skills so that they can customize their PvP-only characters more.
(Hint: Don't play a primary Assassin if you plan to play a lot with other people; there seems to be some sort of bias against them. People just won't take you into a group.)
Personally, I have a Ranger character from Prophecies (1st campaign) and a Ritualist from Factions. I got kind of tired of playing the Ranger, so my Ritualist has become my primary character at the moment.
You can't choose character races. Yet. Who knows what may come up in later campaigns.
If you're having trouble deciding on which campaign to pick up, I'd say that Prophecies is much slower-paced, where you and your group can take as much time as you want for most of the missions. Factions has an emphasis on efficiency, and so getting bonuses on missions often relies on speed. Factions has a nicer newbie area (the island) in my opinion; it offers more tutorials for the new player. Just make sure you hit the level cap (20) before you leave the island, it makes the rest of the game more manageable.
Sorry I'm blabbing a lot because I'm gonna be away from the game for a couple of weeks. Withdrawal issues. | 
