The “Space Race” specifically was the post-Sputnik competitive efforts by the US and the USSR to demonstrate their rocket capability, using their respective cadres of former Nazi rocket scientists.
Rocket research had received somewhat short shrift in the US until Sputnik, carried out by small groups of researchers without a (comparitively) great deal of government backing. Goddard was rejected, von Karman’s group was ignored, and Werner von Braun had to constantly BEG for money.
The real message of Sputnik was :“We can build a rocket with sufficient power and control to place a satellite in a specific orbit hundreds of miles above the earth, and by extension, we can place a rocket with ANYTHING (bombs, troops, etc.) on it ANYWHERE we want to.”
The Cold War sense of the world was that if the US did not show itself to be a formidable opponent of the USSR, there would be little reason for smaller nations not to allow themselves to succumb to Communist rule.
So the Space Race, which really began only AFTER Sputnik, was about showing what could be done with rockets, as a posturing move to suggest potential military capability, although all launches had benefits to other areas, as well. The USSR placed the first man in space, but the Americans created a string of successful space probes and were the only ones to complete a manned Moon mission (following several unmanned missions from BOTH sides).
After that was over, the two powers diverged in their capabilities, the US having more success with their shuttles and probes, the USSR having more success with their manned space stations.
The combination of these successes was to have been the crowning glory if the ISS, but neither technology has turned out be as advertised: the aging shuttle fleet is falling apart , and the MIR turned out on closer inspection to a rather shoddy construction effort.