Voldemort used a Rudimentary Body Potion to return to life as a shriveled, scaly man-child horror.
Later on, Voldemort has access to a Regenerative potion to bring give himself a true form, but afterward, he looks very similar to a snake.
Ha, ha, you look like a snake now vold-y!
But, that just made him look scarier, which just brought more fear to him so I guess it worked for him in the end!
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, a 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.