Titan was discovered on March 25, 1655 by the Dutch astronomer Christiaan Huygens. Huygens was inspired by Galileo's discovery of Jupiter's four largest moons in 1610 and his improvements in telescope technology. Christiaan, with the help of his brother Constantijn Huygens, Jr., began building telescopes around 1650 and discovered the first observed moon orbiting Saturn with one of the telescopes they built.
The name Titan, and the names of all seven satellites of Saturn then known, came from John Herschel (son of William Herschel, discoverer of Mimas and Enceladus) in his 1847 publication Results of Astronomical Observations Made at the Cape of Good Hope. He suggested the names of the mythological Titans (Ancient Greek: Τῑτάν), brothers and sisters of Cronus, the Greek Saturn. In Greek mythology, the Titans were a race of powerful deities, descendants of Gaia and Uranus, that ruled during the legendary Golden Age.
Note that it took over 200 years of squabbling to get a name on this moon. The 'sixth known natural satellite' is also telling as up to 1655 only the earth and Jupiter had know moons orbiting them. It took Huygens using the most powerful telescope in the world to see Titan at all; now you can see it with a pair of 10x50 binoculars. Huygens was also the first to realize that Saturn had a ring system, and correctly identified what Galileo called "ears" 50 years prior.