Simply put, all of the above – depending on where you're from and/or whether you'd consider yourself a die-hard pepper enthusiast.
Many in the UK and Europe refer to them as Chilli Peppers, whereas the majority of the US seems to prefer the americanized Chili Peppers. In Mexico, the Southwest US and some parts of Canada a spicy pepper is generally a Chile Pepper. It's believed early immigrants to New Mexico had converted the Nahuatl chilli to chile.
In Spanish, a spicy pepper is referred to as a chile, which we feel should factor into the discussion as it's commonly believed that all spicy peppers throughout the world can trace their lineage back to Latin America. Additionally, it was the Spanish and Portuguese explorers and traders that had initiated the early migration of spicy peppers from the New World back to many of the cultures across the Old World.
To further complicate matters, there's that well-known meaty dish called chili, which was derived from chili con carne or "meat with chile". Then, there's Chile Powder versus Chili Powder. Chile Powder is made from pure dried peppers, usually with no additives. Whereas Chili Powder contains dried peppers as well as a blend of other spices, which vary depending on the brand/variety of the powder.
To eliminate confusion, provide consistency, and to pay homage to the heritage of spicy peppers, most pepper purists prefer Chile Pepper as the proper spelling for spicy peppers, while dedicating the term chili to refer to the dish. Taking a quick look at the spelling used among industry authorities and publications – The Chile Pepper Institute, Chile Pepper Festival, Hatch Chile Festival, Chile Pepper Food Festival, The Great Chile Book and The Whole Chile Pepper Book – provides further support for Chile Peppers being the preferred spelling.