Indo-European is the most wide-spread and extensively studied language family in the world. The discovery in the 19th Century that Sanskrit was related to Greek and Latin led to the founding of the science of historical linguistics, and contributed to the large part of the science of linguistics in general. The language family encompasses roughly all the languages of Europe (excepting mainly Basque, an isolate, and Finnish and Hungarian, which belong to the Uralic family), much of the languages between Armenia and India, as well as some older extinct languages, such as Hittite. The Indo-Europeans are believe to have lived somewhere in the Caucasus area, and to have spoken a language scientists refer to as Proto-Indo-European. These are the invading Aryans of Indian history, as well as the invaders from the North in many a Mesopotamian account. Here is the general breakdown of this large family, and although it does not include every single language (there are roughly 450 known), it includes all of the better-known examples:
A star indicates that the language is, for all intents and purposes, kaput. If you know of any others that you think should be added to this list, please let me know.