The Space Race was a competition of sorts between the United States and the Soviet Union that started in 1955 and led to the first artificial satellites being sent into orbit, the first humans being sent into space, and the first humans landing on the Moon. The USA was the first to announce intentions to create a satellite, and the race began when the USSR made a similar announcement just four days later, on August 2, 1955. The two large countries were already in an arms race thanks to their simultaneous discovery of German missile technology at the end of World War II, and there is no doubt that the sense of rivalry stemming from that fact contributed to both the Space Race and the Cold War.
As it happened, the USSR launched its satellite, Sputnik 1, first. They also were the first to successfully send humans into space, meaning all of the pressure was on the USA to get to the moon first. This was accomplished in 1969 with the Apollo 11 mission, and after that the Space Race (along with the Cold War) slowly fizzled out.