An excellent question, one that's debated by historians at great length. Who were they? Where did they come from? The Hunnic Empire of Attila, according to current theory, was basically a conglomeration of various people picked up along the way, extremely few of them genuinely "Hun." (Some ancient historians, such as Ammianus, thought they were originally nomads from the Eurasian plain, and this very well might be true, but we don't have any specifics about which we can be sure.) The Roman historian Priscus, who was an ambassador to the Huns, tells of meeting a fellow there who looked as Hunnic as anyone but spoke perfect Greek and turned out to be a Roman who'd been captured and proved himself in battle. Now he was a Hun. We don't even know what language the original Huns spoke; they wrote nothing themselves, and it seems that their empire ended up speaking Germanic. (The name Attila is Germanic, for instance.)

The Empire was put together via conquest: they'd capture a group, and offer to let them join in. If that group did, they got full rights... and a share of the booty when the now-larger war band moved on to the next tribe. Attila himself became king after his uncle, King Rugilia, died, and maintained his position by virtue of being extremely successful. Which is also why we know a relatively great deal about him. Previous Hun wars had been fought against tribes which were outside of the Roman Empire, so the Romans didn't pay all that much attention to the Huns who were prompting many of those tribes to pack up and run across the border into Rome. They were too busy fighting those fleeing Germans. Attila threatened the Empire directly, so they paid attention.

What happened to them? Apparently, after Attila died there was no one left who was strong enough to hold the empire together, and those disparate people fought a huge battle among themselves (The Battle of Nedao, 454) and then just sort of drifted back into being their component parts. Given that there were so few ethnic Huns (or whatever you want to call them) in the first place, they left almost no trace. Jordanes says they fled back to the Black Sea, but was writing a lot later.

And thus the Huns remain a bit of a mystery.

Sources: Peter Heather, Empires and Barbarians Ammianus Priscus Jordanes

This is highly upvoted and six hours in with no answer, so I will give to you the extent of my knowledge of the Huns. Their name was given by Tacitus, who (IIRC) mentions that they lived east of the Volga River. In the two centuries since then they migrated quite a way, fighting in Scythia and elsewhere. More outdated sources will often reference de Guignes' old obsession that the Huns were actually the Xiongnu (who are an interesting story all on their own) but it's really unsubstantiated, and frankly, demeaning to the Huns.

The Huns were, without any doubt, a group of nomads from the steppe. A great many people have lived out on the steppe a great long time - the book on that topic, as far as I'm concerned, is Khodarkovsky's Russia's Steppe Frontier, which is unique in the voice it gives the nomads and their rich history. While 'nomad raiders from the steppe' might not sound prestigious, one must remember that nearly a millenium later, another group of nomads would conquer much of the continent.

Unlike the Mongols, we haven't the faintest idea why they came out from the steppe. As far as I understand, it's generally thought the Huns never had a strong central leadership, but instead came to pillage en masse - Jordanes does mention a leader but Jordanes was also notorious for writing legend as much as history, and Ammianus (kinda) said they seemed to have no ruler.

Settled peoples have been raided by nomadic people forever (or at least as far back as we know). That a potent warlord like Atilla could draw enough killers to his cause is not surprising, either, because that is effectively the story of civilization itself. The Hunnic invasion is just incredible in scale. The Huns came incredibly far west and in such number. There were enough of them that hired Hunnic mercenaries fought bands of Hunnic raiders.

This is truly the mystery. The Huns were not a few bands on the periphery, they were rampaging in a terrific and impressive way. But we do not know of any special motivation they had for this behavior.

Where did they go? Well, likely nowhere. Atilla drew fighters from the territories he raided. His own steppe nomads would be indistinguishable from all the rest of the Turkic peoples around the Black Sea. His Gothic troops would be indistinguishable from the Gothic majority of his Gothic territories. Bands could have ridden home through the European countryside without much fanfare. The dispersion of the Mongol Empire is probably a good comparison to make - Golden Horde and Crimean Mongols were basically absorbed into the local peoples and made such little impact that they're like ghosts in the historical record.

While it is often described as the Hunnic Empire, it was not an Empire and only Hunnic in that it was Atilla's. It's often hard to say how much the people who were called 'Huns' were really Huns or were just barbarians doing what barbarians had always done. They were not a monolith; they were a great many people of which little was recorded about. But in truth that is the story of so many nomads who harried the civilized world for millennia.