Although the character, Nightcrawler, was adapted very closely from the original, comic book character several differences were made to make the character easier to portray on screen and to fit into the movie's storyline.
Nightcrawler in the comics looks very much the same as he did in the movie.
The most notable difference would be his skin. The comic book Nightcrawler does not have ritualistic scars, instead he is covered with a fine, blue fur that has earned him the nickname "Fuzzy Elf."
Also his eyes are completely yellow with no visible whites or pupils.
His teeth are more human looking than in the movie, but he does have pronounced, fang-like canine teeth.
His tail is smoother with no ridges, and only a flat, triangular spade at the end.
The story of how Nightcrawler came to join the X-Men was changed from the original to fit the movie. Originally, Nightcrawler joined the X-Men at the same time as Storm and Wolverine. He was recruited by Charles Xavier who used his mental powers to stop an angry mob from killing him. He has fought as one of the X-Men for many years, and even for a time lead a separate team, Excalibur, a European version of the X-Men.
Note: This is for the benefit of people who have seen the movie(s) but are not at all familiar with the comic.