The nose of Tom Riddle (as a student) was fine, as I have seen in the movies. But, Voldemort's nose doesn't look like a normal human nose. What really happened?
It was degraded by Dark Magic.
Slytherincess is quite right to mention the meeting between Voldemort and Dumbledore as a midpoint between the youthful Riddle and the You-Know-Who that subsequently emerged.
Voldemort had entered the room. His features were not those Harry had seen emerge from the great stone cauldron almost two years before; they were not as snake-like, the eyes were not yet scarlet, the face not yet masklike, and yet he was no longer handsome Tom Riddle. It was as though his features had been burned and blurred; they were waxy and oddly distorted, and the whites of his eyes now had a permanently bloody look, though the pupils were not yet the slits that Harry knew they would become.
(Half-Blood Prince, Chapter 20, Lord Voldemort's Request).
This is a Voldemort in transition. He was, as Britney Spears might put it, not a girl, not yet a woman. Or rather not Tom Riddle, not yet You-Know-Who. Alongside the rest of his features, his nose had undergone some adjustments. Nevertheless, it wasn't yet the flattened, slit-nostrilled monstrosity it would go on to become.
(Quickly, to deal with the point about the movies, the film-Voldemort has almost no nose at all. This is arguably a diversion from the books where the nose is merely described as being "flat as a snake's, with slits for nostrils". Having a flattened nose is not the same as having no nose at all...).
The reason for Voldemort's physical deformity is that he has spent much of the last decade experimenting with Dark Magic. Voldemort's long-term aim was immortality but the Riddle who soaked up magical knowledge like a sponge was I think also immensely curious to uncover the darkest secrets that magic had to hold. He would've had no qualms about inventing his own hideous spells or performing magic on himself that no other wizard would even contemplate. Voldemort was experimenting with Dark Magic that was uncharted territory in both its complexity and its depravity.
"I, who have gone further than anybody along the path that leads to immortality. You know my goal - to conquer death. And now, I was tested, and it appeared that one or more of my experiments had worked...for I had not been killed, though the curse should have done it."
(Goblet of Fire, Chapter 33, The Death Eaters).
"I think you must know that I have seen and done much since I left this place. I could show and tell your students things they can gain from no other wizard."
Dumbledore considered Voldemort over the top of his own goblet for a while before speaking.
"Yes, I certainly do know that you have seen and done much since leaving us," he said quietly. "Rumours of your doings have reached your old school, Tom. I should be sorry to believe half of them."
[...] "I have experimented; I have pushed the boundaries of magic further, perhaps, than they have ever been pushed -"
(Half-Blood Prince, Chapter 20, Lord Voldemort's Request).
Clearly creating Horcruxes was part of this process of experimentation. However, the phrase "one or more" certainly implies that Voldemort was experimenting with several different techniques of gaining immortality. Horcruxes were just one of his options. Remember that Voldemort was a brilliant wizard who mastered magic that perhaps nobody else had even discovered. He had to accumulate that knowledge somehow, and these experiments appear to be a key part of the process of his magical development. What forms of magic Voldemort was experimenting with are never made clear - but I think that we can justly assume that they were all dark and terrible. Voldemort is therefore a kind of Frankenstein, a dreadful experiment and a perversion against nature. I don't think we can say with any clarity what exact spell ruined his nose (or eyes, face or voice). Just that a series of experiments with Dark Magic periodically deformed his appearance.
Dumbledore at least seemed to think that the soul-mutilation from creating the Horcruxes had a large part to play.
"Yet it fitted: Lord Voldemort had seemed to grow less human with the passing years, and the transformation he had undergone seemed to me to be only explicable if his soul was mutilated beyond the realms of what we might call usual evil..."
(Half-Blood Prince, Chapter 23, Horcruxes).
Although, according to Mad-Eye Moody, malfunctioning wands can cause nasty accidents which can result in physical deformities. Who knows what other body parts Voldemort lost in his experiments with Dark Magic?
"Don't put your wand there, boy!" roared Moody. "What if it ignited? Better wizards than you have lost buttocks, you know!"
(Order of the Phoenix, Chapter 3, The Advance Guard).
Voldemort's appearance became more snake-like as he delved deeper into Dark Magic, and his face became more serpentine, which would presumably explain the flattening of his nose. Mention of Voldemort's serpentine appearance is mentioned in every book. It's theorized that each murder Voldemort committed, the more distorted his face became. There is a scene where Dumbledore shows Harry a memory of Voldemort after he resurfaced after being gone for ten years. He came to hide the Ravenclaw Horcrux and to ask Dumbledore, who was newly appointed headmaster, for the Defence Against the Dark Arts position, and his appearance is drastically changed (this would be been around circa 1956) from when he had gone to work at Borgin and Burkes ten years prior. It's fair to speculate Voldemort was deeply involved in Dark Magic; Dumbledore notes that he hopes "half of what he's heard (about Voldemort's Dark Arts activities) isn't true."
Regarding the movies, super short answer: CGI removed Voldemort's nose. There's a segment in the extra scenes where Voldemort's nose is discussed extensively. Ralph Fiennes was very opposed to the flat nose and had to be convinced of the scary aesthetic value of that choice.
Having read the Harry Potter Series, it is safe to say that J.K. Rowling makes it perfectly clear why Voldemort's face is serpentine. As his soul is being torn apart (to make Horcruxes) his appearance is little by little becoming more snakelike. At his level of practicing the Dark Arts, evil curses that draw upon one's energy might have have had an effect on his face also. Being an avid Parseltongue and Slytherin might have also had an effect. Nowhere in the books does Rowling directly address if all of Salazar Slytherin's Dark-Arts-practicing descendants take on a serpentine appearance over time. Being as Voldemort was the wizard who performed Avada-Kedavra (the Killing Curse) the most, no other wizard (even a member of the Death Eaters) could've adopted the snakelike appearance. All of these combined are what most likely cause Lord Voldemort to appear snakelike.
Apart from the more elegant answer that it was a result of losing pieces of his soul, let us also take into account that: "the Weasley twins were punished for bewitching several snowballs so that they followed Quirrell around, bouncing off the back of his turban."
If I can recall correctly, it was because of the potion he made to come back to life. In the first book, it doesn't say that he doesn't have a nose, which is why I believe this. Voldemort drinks a potion, which is brewed by Wormtail and is taken periodically, which consists of unicorn blood and Nagini's venom. The vemon which helped keep Voldemort alive would have made him appear more snakelike.
It says in the books that the snake-like eyes and nose are because of the fact that he made a snake into a horcrux... Making the first 6 horcrux's make his eyes red, his expressions ugly, the whites of his eyes red, and most likely caused the loss of hair and caveman brow but the slits of his eyes and nose are because of sharing his soul with Nagini.