The Indo-European languages spread over most of Europe and large chunks of Asia at some point in pre-history. The discovery of relationship between European languages and Sanskrit was a major catalyst for the development of the science of linguistics. Most of what we know about historical linguistics suggests that this kind of language relationship could only have come through something like conquest and replacement of other speakers. The most popular scholarly hypothesis is the Kurgan hypothesis - that the original Indo-Europeans were nomads of the Pontic steppes, now Eastern Ukraine and Southern Russia, who domesticated the horse and conquered much of the world.
There is a fair amount of archeological evidence to back up this idea, but some crucial uncertainties remain. One problem is getting the inferred dates to line up. Another is that it's hard to trace the gene flow that accompanied the putative expansion. India, with its intricate and complex genetic history, presents special problems. Many assert that the gene flow history of India is not compatible with a major incursion from the steppes at the time suggested by the archeological evidence. See, e.g., this link by Arun.
There are a few alternative theories of I-E origins, discussed in the first link above. One, mostly held by Indian nationalists, is the so-called out of India hypothesis, in which the I-E expansion originated in India. This idea is unpopular because it doesn't seem to fit linguistic or archeological evidence. The horse, for example, is central to I-E, but is not native to India, mostly doesn't thrive there, and does not appear in iconography of the Indus Valley Civilization, the civilization in the Indus valley which probably predates the I-E appearance in India.
The genetic evidence is complex and controversial, but at present mostly is centered on the R1a1a variant of the Y - chromosome (male germ line), see map here. Not sure whether more general genetic evidence has anything to add.