There has in history only ever been one space race. Therefore, the more appropriate question is probably “what was the space race.” The space race was a competition in the 1950s and 1960s between the Soviet Union and the United States. This was a competition in which both countries were trying to be the first to achieve various milestones related to space exploration. The two countries wanted to win the space race for two main reasons.
The space race did not have any official beginning. However, it can be said to have begun in 1955. In that year, both the US and the Soviets announced that they would soon be able to launch artificial satellites into orbit of the Earth. The race then heated up dramatically when the Soviets won the race to the first milestone. This occurred when they launched Sputnik, the first artificial satellite, in October of 1957. The Soviets also reached the next milestone first when they put the first human being into space in 1961. The main part of the race that the US won was the race to put a human being on the moon. The US accomplished this in 1969.
The space race was important for two reasons. First, it had military implications. The country that could build better rockets would have an edge in building nuclear missiles. Second, it had ideological implications. Other countries would notice which country was winning the race and would tend to be attracted to that country’s ideology. Thus, winning the space race was seen to be important both for military and for public relations purposes.