The domino theory stated that once Communism began to spread in one country, it would also spread to adjacent countries, and then more and more countries until it took over the world, as one domino knocks over the next, which knocks over the next, until they're all down. On this vision, Communism is something like a virus which spreads on contact. (This isn't as crazy as it sounds; ideas behave like mind-viruses in many respects.)
Most historians think that the domino theory was very important in motivating the decision to fight in the Vietnam War (which was never technically a war, actually; it was considered a "police action" under the law so that it didn't have to be approved by Congress). It definitely featured heavily in the public justifications for the war; most Americans did not consider Vietnam particularly important per se (it was sort of a backwater country many Americans had never even heard of), but the fear that losing Vietnam might mean losing the rest of Southeast Asia or even spreading Chinese Communism into Japan and India was highly motivating.
There are some, however, who believe that the domino theory was not the primary motivation for the Vietnam War and was more like an excuse the government used to garner support. This is not a very popular view among mainstream historians.