How was the Hubble Telescope transported into orbit?

HST was transported into orbit aboard Space Shuttle Discovery (mission STS-31) in April 1990. It was scheduled to be orbited earlier, but the shutdown of the Shuttle program following the Challenger accident delayed its deployment, not to mention increasing its maintenance cost because it sat on the ground (about 1 million per day). Astronauts successfully deployed and boosted HST into its orbit altitude of 569nm.

Though any of the Shuttles can perform an HST Servicing Mission, Shuttles Discovery and Endeavour are usually preferred due to their larger payload bays (we can put more stuff in it).

To date, HST has been in orbit and providing science data for over 20 years.