The first telescope is generally credited to Hans Lippershey, a Dutch spectacle maker, who applied for a patent for the device in 1608. While he may not have been the only one working on the idea at the time, his design was the first officially recorded and described.
Shortly after, in 1609, Galileo Galilei heard about the invention and built his own improved version. Galileo was the first to use a telescope for astronomical purposes, observing the moons of Jupiter, the phases of Venus, and craters on the Moon—revolutionizing astronomy.
So while Lippershey invented it, Galileo made it famous.