Padan Fain was a Lugard peddler, who moonlighted as a Friend of the Dark. When it came time for the Dragon to be reborn, Fain was taken to Shayol Ghul and made into the Dark One's Hound, to search out the Dragon Reborn. He followed the boys to Shadar Logoth and had a run-in with Mordeth. Mordeth tried to devour Fain's soul, but couldn't, because of the hold the DO had on him. So Fain became part Mordeth, part renegade minion of the DO. This is basically what the books tell us.
What is he up to now? Basically, playing the part of picador to Rand's bull, popping up all over the place, poking, prodding, inflaming, and generally causing trouble.
He went to the Fortress of the Light and the White Tower to sow seeds of dissension, and make sure Pedron Niall and Elaida would never join Rand. He instigated a failed assassination of Rand by his ex-WCs in Caemlyn. He might also have been responsible for the attack on the Brown AS in Caemlyn which ended up driving a rift between Rand and the Salidar AS, and sending him into the hands of Elaida's AS in Cairhien, but this could just as easily have been part of some Forsaken's plot, or a plot by the Tower AS and the Shaido to alienate the Salidar AS from Rand (see section 1.4.07).
In ACOS, Fain appeared in the company of Toram Riatin (a Cairhienin rebel), calling himself "Jeraal Mordeth", and gave Rand his second unHealable wound. He may or may not have been responsible for the evil people-eating fog that appeared in the rebel camp as well (though it was more likely another "bubble of evil").
Finally, in WH, Fain kills off Kisman, Torval, and Gedwyn in Far Madding, to keep them from getting to Rand before he does. We originally thought that the bizarre appearance of the recently dead Torval and Gedwyn walking up the stairs at the inn [WH: 33, Blue Carp Street, 615] was further evidence of Fain's powers; however, it's been suggested that this was actually an early occurence of the ghost phenomenon seen in COT (see section 1.6.3).
As far as we know, Fain still has his pet Fade.
Roy Navarre and Tony Z came up with a loony theory that Fain is actually the avatar of the DO. Roy says: "First, if you check the glossary, you will see that the DO is described as the source of all evil. Hence Mashadar must flow from the DO or the glossary is wrong. (Note that that last option has been known to happen.) Next, Myself and Tony Z presented detailed evidence suggesting that Fain is the avatar of the DO. With each broken seal, Fain gets stronger. Thus, the DO has been in our midst all this time but we just didn't know it. At first only a trace of him in Fain, but growing stronger and stronger until now his presence in Fain seems unmistakable."
Eric Ebinger counters: "Fain no longer exists. Padan Fain was summoned to Shayol Ghul, was broken and reformed into a bloodhound for the DO, as part of which he was imprinted by the DO. This happened twice at Shayol Ghul and once in a dream. Padan Fain/DO bloodhound went to Shadar Logoth and fell prey to Mordeth. Normally, Mordeth would just destroy the existing "soul/personality", but Padan Fain's having been "remade" by the DO seems to have changed things sufficiently so that there was a slow gradual merging of all of the different personalities (Fain/Mordeth/DO's imprint). The most accurate term for the combination is the name that he took: Ordeith. Over time, the Mordeth portion has gained more and more control over the gestalt. The DO's imprint has given Ordeith the unreasoning hatred of Rand, Perrin, and Mat. There doesn't seem to be much of anything of Padan Fain left. As the Mordeth fragment has gained more complete control of the gestalt, Ordeith has increased in power. The seeming relationship between the breaking of the Seals and Ordeith's power is due only to the fact that as time passes Ordeith gets stronger and as time passes the Seals break. The same relationship is evident with Rand, Perrin, Mat, Elayne, Egwene, Aviendha and Nynaeve." Note that Fain is now calling himself "Mordeth," which suggests that the Mordeth part is dominating, which makes it very unlikely that Fain is the DO's avatar.
Furthermore, as John Novak states: "If Fain is now an embodiment of the Dark One, why in Hell was Slayer hunting him down as a renegade in TSR? Does the Dark One like being hunted by his own servants?"
Finally, it's pretty apparent that if anybody in these books is the Dark One's avatar, it's Shaidar Haran (see section 1.4.05).
Some people have suggested that the Mordeth aspect of Fain (which, as we have noted, now seems to be the most dominant part of him) will be weakened or even killed as a result of Rand blowing up Shadar Logoth.
This doesn't seem likely, though. For one thing, the wound Fain gave Rand with the SL dagger is still there, unchanged [WH: 35, With the Choedan Kal, 655], [COT: 24, A Strengthening Storm, 546]. Ben Goodman points out, "Mordeth more or less brought Shadar Logoth into being. His binding with Fain made him independent from it although he could draw power from things connected to it like the dagger. The dagger itself can be seen as a part of Shadar Logoth that survived. It was the evil rather than the location that gave Fain and the dagger their power. I don't think that there were invisible cords linking Fain and the dagger to Shadar Logoth so that when its evil is consumed by the Taint, their evil is consumed too."
It's possible that Rand's wound from the dagger cannot be Healed until the dagger itself (and possibly Fain along with it) is destroyed [Maccabeus Epimanes]. Since the wound from the dagger is still the same, it seems safe to assume both the dagger and Mordeth/Fain are (relatively) unaffected by SL's destruction.