They essentially both mean 'what', but they are used beside different sounds. Nani --> used with o and other vowels Nan ---> used with d, t, n, and counters Ex. Nani o suru? Nan desu ka.

nani is 'what' or 'what kind'. you can say 'nani?' by itself nan is more of how much, how many the above are basic rules but does not apply to every situation

nani is "what?" nan is the short term for nani. (as far as i know lol.) ex. nandesuka? sometimes they talk fast and they omit some vowels.