It's generally agreed that the Titans are on a self-imposed quest to bring order to the chaos of the universe. (Well, if it's not self-imposed, we as yet have no idea who is powerful enough to tell the Titans to do chores, at any rate) As part of this quest, some or all of the Titans visited Azeroth in the dim past when it was ruled by the Old Gods, unpleasant sorts who seem to like the universe, or at least Azeroth, just fine all nice and chaotic and who took umbrage at the Titans just showing up and demanding that they clean up their room. This led to a, if you'll pardon the expression, titanic war between them that only ended after the Titans had kicked the Elemental Lords themselves up and down the block and jammed them into their prison in the Elemental Plane and then rolled up their sleeves and began the arduous task of finding asses in the wriggling mounds of eyes and tentacles that the Old Gods lounge around in, and then kicking those asses and sticking their defeated enemies into a prison deep within the earth of Azeroth itself, except for the one they killed in Darkshore and C'Thun, who ended up half-dead in Silithus, possibly after having killed a Titan himself.
Having done this, the Titans proceeded to do what they apparently do all over the universe and straightened the mess up. They created the oceans and shaped the land, calmed the furious storms caused by the formerly-amok elementals, and generally established the ebb and flow of life as the mortal races would come to know it. They appointed the five Aspects to shepherd the world once they were gone and then left, positive that nothing could possibly go wrong. They may have even said that as they left, because as powerful as they are they don't seem to have a really good grasp on drama and didn't realize that saying 'Nothing could possibly go wrong with our creation' practically guaranteed that something would go wrong. For starters, it's unclear if the Titans made the Well of Eternity or not, and even if they did, they don't seem to have foreseen how it would change their creations and alter the face of Azeroth - the War of the Ancients being just one example of how the Well's magic drastically altered what one could see as the Titan's 'design' for Azeroth. The Titans also seem to have counted the Old Gods out a little prematurely, as the corruption of their aspect of Earth, Neltharion, would indicate. Finally, the Titans seem either unwilling or unable to do much of anything about the Burning Legion. Of course, that could be because they are in fact engaged in combat with the Legion across the infinite Great Dark, but from the mortal perspective the Titans seem to be absentee landlords, the kind who have a real jerk of a nephew who keeps everyone else in the building up all night raising the dead and destroying Dalaran.
The Titans appear to be ruled by a collection of powerful individuals known as the Pantheon. We don't know much about these guys either, except that they directly gave power to the Aspects and that each of them has a sphere of influence, sort of like a cosmic hobby, that they specialize in. The big names among the Pantheon are Aman'Thul, Eonar, Khaz'Goroth and Norgannon. Sargeras doesn't really count anymore, and both Golganneth and Aggramar seem like younger Titans who rose to positions of more prominence once Sargeras went nutty and decided to destroy everything in the universe.
Of these Titans, Aman'Thul is basically the leader of the whole shebang, known as the 'Highfather' and he's the guy who gave the aspect Nozdormu his powers over time. He's also Sargeras' brother. What is it with brothers in the Warcraft universe? How is it the only well-adjusted brothers in the whole universe are the Bronzebeards, and one of them is dead and another is constantly crawling around underground losing the bases of artifact level staves inside Old Gods?
Eonar the Lifebinder is the one who empowered Alexstraza and Ysera, and besides being the Titan most concerned with nature and healing, she's also apparently part of the reason that Sargeras got so ticked off at his big bro and went off in a snit to throw the universe's biggest temper tantrum.
Khaz'Goroth is the Titan who likes to make stuff, the forge-wright of the Titanic host, and the guy most directly responsible for the creation of the Earthen and thus for every wipe you ever suffered in Uldaman. He's also the one who thought it would be peachy-keen to empower Neltharion as the Earth-Warder without warning him to not listen to the mutterings of the Old Gods or else he'd go coo-coo for chaos puffs. So we can also blame him for Deathwing, too. Thanks, Khaz'Goroth. I wouldn't be surprised to find out Khaz'Goroth also called Sargeras names and gave him a wedgie or something, but no one's reported any Titanic wet willies to explain the creation of the Burning Legion as yet.
Finally of the big four remaining Titans we have Norgannon. Not only is Norgannon the home of my alliance characters and a lot of my old guildmates (Hi Mazrimtaim, Tyranna and Sedal) but he's also the one who chose Malygos as Aspect of Magic. You can find his Discs in Uldaman and take them to Uldum in Tanaris, whereupon you learn quite a bit about the Earthen, the Dwaves and the creation of Azeroth, and hopefully someday there will be another instance or two continuing the questline. Norgannon is basically the biggest expert on Arcane Magic you're ever likely to even hear about, in addition to his general expertise on lore and history. If Norgannon doesn't know it, he's working on learning it.
As to what the Titans are up to nowadays, no one really knows. They could be off blissfully unaware of what's going on back on Azeroth, ordering new worlds to their specifications. They could be fighting the Burning Legion, they could be ordering Tyrael to destroy the Worldstone for all I know. But even though you don't see them around much anymore, any time you run into a dragon, a giant, or some ancient ruin or another, fight Ragnaros or another Elemental Lord, or deal with any Old Gods, you're running smack into the legacy of the Titans, and that's without even considering how there wouldn't even be a Burning Legion without them.
Next week Elizabeth will be back to rock your socks (or whatever you wear on your feet) off with more Dragons, unless she does something else that will be equally awesome. Then I'm back, and unless things change mightily, I'm going to focus on our least favorite former Titan, the guy who caused all the trouble with demons we're currently dealing with in Outland, our boy Sargeras himself.