Chile, or "chile pepper" is the adopted American English term for the fruit of the Capsicum plant. The plant probably originated in Perú, where it is called ají (pronounced ah-HEE). Thanks to Inca traders, the plant made its way to North America long before Columbus arrived and "discovered" it. The Aztecs called the plant chil in their Nahuatl language, and from this we get the Mexican term chile.
Columbus' voyages were motivated by the lucrative spice trade. Spices were fabulously expensive in Europe, and one of the most prized was Piper nigrum, or black pepper, from India. Christopher Columbus was a navigator, but also a marketeer (his voyages needed financing), so it made perfect sense to him to call the people he found indios and the pungent Capsicum plant pimiento (pepper). Although we know better today, these two gross misnomers hang on stubbornly. We still say "hot pepper" or "chile pepper" when referring to chiles, and in Spain, all chile spices are still called pimentón. (In the British Isles, all pungent spices are culinarily classified under the Indian word curry.)
"Chili" is another spelling, but this term is more commonly applied to the Southwestern dish with meat and beans.
(The name of the country of Chile is also an Amerindian word, but is Aymará, unrelated to the Nahuatl, and comes from chilli, meaning "land at the edge of the world." Thus the chile plant has no connection whatsoever to the country of Chile.)
Thanks to the Spanish traders, today chiles are grown all over the world (the best in the hotter climates). Half of the over 200 varieties of chile are indigenous to México. The most common species of Capsicum consumed in the U.S. and México is Capsicum annuum, and is relatively unknown in the Americas south of the equator. Ají (Capsicum baccatum), not chile, is the species of Capsicum consumed in most of South America. (Capsicum baccatum is believed to be the ancestor of Capsicum annuum.)
Ají is well known in Chile, while chile is not. The cultivar ají verde (green ají) is the distinct Chilean species of ají, also called ají merquén by the native Mapuches. It is most often eaten when fresh and green, when it is mildest and most flavorful, in salads such as the Ensalada Chilena, or pebres, or as a condiment for soups or other hot dishes. In cooking, the more ripened yellow merquén may be used, while the fully ripened red merquén, also called cacho de cabra (kid's horn) is almost too pungent to eat, except when used judiciously as a seasoning in criollo (Creole) dishes or Spanish pilpil. Some imported chile spices are known in Chile, such as cayenne pepper and paprika, but they are misnamed ají de cayenne and ají de color. In Chile, paprika is made from the sweet ají pimiento porron, and has a distinct flavor from Hungarian paprika.
Chileans, whether because of isolation or their strong Basque-European influence, have not acquired a fondness for pungent foods, in contrast with their Peruvian and Bolivian neighbors. For this reason, many visitors have described Chilean cuisine as bland, except in the countryside, far from the cosmopolitan capitol, where criollo cooking abounds and flavorings enhance the humility of the ingredients.
For some reason, ají is not commercially grown in the United States, the world's sixth-largest producer of chile (after China, Turkey, México, Nigeria, and Spain).