Sputnik is the name of a series of artificial satellites built by the Soviet Union in the late 1950s. Generally when people use the term they mean Sputnik 1, which was the first satellite launched into space on 4 October 1957.
In Russian, "sputnik" means someone or something that follows you or something or somebody else. The key word is "follow" here. It's now a catch-on term for a satellite, artificial or natural (moon).