Each variety of chile has a subtly different flavor, but generally the kind to use is determined by how spicy you want the dish to be; spicier dishes need hotter peppers, otherwise you end up with a dish dominated by the peppers. For this reason, most people sort chiles by their spiciness, measured in Scoville Heat Units (SHU). The exact same papers can be used for Mexican and Indian cooking, as long as you want similar spiciness.
To give options, I like to keep 3 kinds of dried peppers: a mild paprika, a standard hot pepper, and a ground, smoked one. The sweet paprika is for savor, the hot pepper to add spiciness, and ground, smoked peppers bring that distinctive smoky flavor to dishes. I usually use de Arbol pepper flakes (about 30k SHU) to add heat, and smoked hot paprika for smoked stuff. When I have a dish that showcases the flavor of a specific pepper, it's time to get fresh ones from the store.
For general purpose cooking, many people use mild paprika to add sweet, smoky flavors to Italian, French, Spanish, and Balkan cuisines. Generally these are of the Capsicum annuum species.
Indian, Szechuan Chinese, and Central American cuisine normally use moderately hot peppers, in the 10-30k SHU heat range. Cayenne, tabasco, and de Arbol peppers are good examples. Thailand (and some parts of India) use very hot peppers up to 100k Scoville, called Bird's Eye or Thai chiles. These peppers may be either Capsicum annuum (normally milder) or Capsicum frutescens (normally hotter).
Finally, Mexico and northeastern India use the hottest peppers of the Capsicum chinense species, including habaneros and the naga jolokia, which go from 100k SHU to 750k SHU.
Mexico is a special case, because many varieties of pepper are mixed to get the desired flavor. Everything from bell peppers to habaneros gets regular use, and may be used smoked or dried.