The Space Race occurred between 1955 and 1972 at a time when there was almost war (called the Cold War) between the United States and the Soviet Union. The last frontier was space, and the two countries began a competition to see who could get there first. It was like winning a battle or a war.
Both sides knew it was good for the morale of their people, their standing in the world, and also because they knew that so much of the future would be controlled by whoever was able to get into space.
Interestingly, the Germans had been far ahead of both the Soviet Union and the United States, launching missiles during World War 2. So at the end of the war, both the Soviet Union and the United States were keen to take over the work of the Germans. Some went to the Soviet Union, others to America. As a result, both started the Space Race with the same German know how.
The easiest thing to do was to launch a satellite to orbit the Earth. This technology is used today by the thousands of satellites orbiting above us to give communications, GPS and so on.
More difficult would be unmanned probes to the Moon, Venus, Mars, and beyond, and most difficult of all would be human spaceflight either around the Earth or to the Moon.
On October 4, 1957, the Soviet Union launched the world's first satellite. It was called Sputnik 1.
This was a great shock to the Americans, whose national pride was dented. As a result President Kennedy announced the missions that would land people on the Moon. This was achieved on July 20, 1969 with the Apollo 11 mission.
Many new advances in communications, materials and electronics came as a result of the Space Race. It was hugely expensive, but its achievements are seen in many of the things we use today.