At the end of the movie Harry Potter and the Deathly Hallows, Part 2:
Voldemort and Harry had their final face off. When Harry won, Voldemort disintegrated.
Why did this happen?
Note that I'm looking for answers consistent with the movies. Information from the books that explain what happened while remaining consistent with the movies is acceptable.
The old answer based on the books can be found further down. But as GreenMat insists on an answer from the movies I have to say: There is NO logical explanation in the movies for what happens to Voldemort. Neither does the disintegration happen in any other instance of a duel nor is there any spell in the movies that can cause something like this.
The closest thing to the disintegration we get is the effect on Quirrell when touching Harry in the first movie, where something very similar happens.
So a movie- explanation MIGHT be, that it is the protection from his mother that lets the bad guys disintegrate. But still this is inconsistent with the fourth movie, where Voldemort -after getting his new body- can touch Harry without disintegrating.
As you can see in 4th, 5th 6th and 7th movie: whenever Voldemort tries to kill Harry he uses the unforgivable curse number one Avada Kedavra.
It always looks the same (green flash) as the spell in the last duel. And Harry always uses Expelliarmus, the red beam, especially in the duel in the fourth movie at the graveyard.
We also see in the movies that Voldemort is able to do magic without actually saying the words (e.g. when he forces Harry to bow by just pointing his wand at him.
These two facts let us assume, that it was Avada Kedavra again -this time unspoken- in the last duel.
WHY the bounced back spell lets Voldemort disintegrate while normally it "just kills" is not explained but could be made to refer to Quirrell in the first movie.
Answer from the book:
The movies are quite inaccurate in a lot of ways, and this is one of them.
In the books there is no difference between casting a spell and duelling. In the 6th book we learn, that experienced wizards can cast spells by just "thinking" them without speaking them out loud. They learn the "unspoken spells" at school.
It is said in the books that casting a spell without saying it loud gives you an advantage especially in duels, as your opponent cannot prepare for the spell as he doesn't hear it.
In the final duel in the book none of the two uses an unspoken spell: Harry casts his favorite spell "Expelliarmus" while Voldemort uses -guess- "Avada Kedavra". As Harry wins Voldemort's spell bounces back to him and kills him (this is NOT normal in a duel but due to the fact of the Elder Wand's true allegiance).
And no, he does NOT disintegrate, this is made for the effect of it only in the movie.
To answer your question: the loser just experiences the spell the winner cast. Of course a Death Eater might kill the loser with another spell after winning, but that is not part of the duel.
The body he re-built was flawed
The spell used to bring him back:
Bone of the father, unknowingly given, you will renew your son! Flesh of the servant, willingly sacrificed, you will revive your master. Blood of the enemy, forcibly taken, you will resurrect your foe.
While he rebuilt body with the spell, his soul was already broken, several times.
You could interpret the soul as the glue that holds a persons body together, and that once Harry and co destroyed the horcruxes, Voldemorts's body couldn't hold itself together, and so he disintegrates.
Voldemort had already cast one major killing curse (in the forest), which knocked him off his feet. Trying another + no more horcruxes was too much for him.
Similar to what Torsten Link has answered above which was explained very well: the synopsis that happened in that part of the 8th movie is just a false information, Voldemort just died, but it is the book which where these movies came from, and the book is considered canon since it has every info you need to know and this was the official scenario of J.K's harry potter series.
In the book, they didn't fly in black smoke, which was performed by Voldemort. And before the last duel of Harry and Voldemort started, they still managed to exchange discussions including the true loyalty of Snape and the Elder Wand.
But this is just a recap of what I have read in the book 7. But then, you asked what happens to someone who loses a duel in the movie. And you cited an example of why did Voldemort disintegrate in the 8th movie. In the book, Voldemort was killed by his own spell.
As for what happens to someone who loses a duel, they lost, or sometimes die if and only AK spell hit him/her or something dangerous spell hit him/her. (NOTE: I am not generalizing each and every scenarios that when someone was hit by a dangerous spell it kills him/her) As for what is the reason why did that happen in the movie, because that scene in the movie makes no sense at all.
Though why exactly the Dark Lord exploded as he did in the movie isn’t explained in the movie itself, a similar thing happens to Gormlaith Gaunt, and in her case it was caused by the Dark magic she’d used reacted with the Pukwudgie venom on the arrow she was shot with.
The old witch had indulged in all manner of Dark magic in an attempt to make herself invincible, and these curses now reacted with the Pukwudgie’s venom, causing her to become as solid and as brittle as coal before shattering into a thousand pieces. The Ollivander wand fell to the ground and burst: all that was left of Gormlaith Gaunt was a pile of smoking dust, a broken stick and a charred dragon heartstring.
- Ilvermorny School of Witchcraft and Wizardry (Pottermore)
This is a known case of a Dark wizard disintegrating upon death. The circumstances are a bit different, since the Dark Lord is killed by a rebounding Killing Curse, not Pukwudgie venom, but Gormlaith’s death seems very similar to how the Dark Lord died in the movie. The Dark Lord also had used a lot of Dark magic to make himself more powerful, including the potion that restored him to a body. He’d used much more Dark magic than most Dark wizards ever had, so it’s entirely possible that a Killing Curse hitting him could cause a different reaction than it normally would.
An in-universe explanation is that while the AKC ordinarily leaves no trace of operation on its victim, other than their addition to the Chicago voting rolls, the same curse when rebounded against its caster can have a very different effect. In the books, Voldemort's body exploded at the first attempt to kill Harry.
