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Before the space shuttles were invented, only way to explore space was using the optical appliances, like telescope. It wasn't very good, as you could look at the space only from one perspective, but it was used from 1609 year.
The first unmanned flight was made by the USSR in 1957, when they placed a satellite by the name of Sputnik-1 onto the earth orbit. But the first flight of a living creature happened around a month later in November of the same year, still by the USSR. It was a dog by the name of Laika and traveled on the Sputnik-2 to the orbit, but died in 5 to 7 hours after the startup. Sattellites are still being placed on the Earth orbit, mainly to serve geolocating and internet-connection purposes.
Manned flight was completed by USSR astronaut Yuri Gagarin in 1961, on the 12th of April on the rocketship Vostok. He travelled around the earth orbit and safely returned to the ground. Humankind flights are not ending even to this day because the space is still very unexplored and full of dangers.
Competition started almost right after Second World War, as the countries, mainly USSR and USA, tried to gain their dominance in the space. They sent their sattellites and spaceships as fast as they could only to gain that status. Nowadays both countries cooperate in their space exploration methods, as the astronauts of both countries are located on the ISS, on the orbit of the Earth.