Both of these phrases were coined during the expansion of the Eastern Bloc during the Cold War. 

The Americans believed that countries in Asia and Eastern Europe were falling 'like dominos' to communism. So in other words as one country succeeded to communism, so too the countries bordering it would follow. And this theory, as far as the Americans could see, was actually happening.

So the US response to this was containment; simply put the US would put as many resources as was necessary into stopping the dominos falling and halting the spread of communism. This included funding anti-communist rebellions and political parties in Eastern Europe. But the most famous example of containment in action was the direct military involvement of the US in the Vietnam War; a demonstration of how far the Americans would go to stop Communism gaining further momentum.