That's a great question! The name of peppers can vary from region to region, hence the different spellings. Depending on where you go, it is either "chili pepper", "chilli pepper", or "chile pepper". Take your pick!
"Chilli" is more commonly used in England and Europe. "Chile" is by far the most commonly used in America's Southwest regions and particularly in New Mexico. "Chile" is actually the Spanish spelling and pronunciation of the word. However, "chili" is far more common in general usage and spelling, an Americanized spelling of the original "Chile" spelling that came out of Mexico.
Die-hard chile pepper lovers argue that "Chile" is the only proper spelling, and that "Chili" refers only to the delicious food with meat, beans and chili spices that you eat out of a bowl, but most people use "Chili" or use both "Chili/Chile" interchangeably. Unless you're from the U.K., then it's probably "Chilli", though they recognize all variations as well.
Either way, all spellings are recognized by the dictionary.