The planet names are derived from Roman and Greek mythology, except for the name Earth which is Germanic and Old English in origin. The five planets easily visible with the unaided eye (Mercury, Venus, Mars, Jupiter, and Saturn) have been observed for all human history as far as we can tell, and they were called different things by different cultures. The Romans named these planets according to their movements and appearence. For example, Venus, the planet that appears the brightest, was named after the Roman goddess of beauty, while the reddish Mars was named after the god of war. These Roman names were adopted by European languages and culture and became standard in science.
When Uranus and Neptune were discovered, there was not an established tradition in place so a few names were considered and used for each planet, until one name became standard. William Herschel, who discovered Uranus, wanted to name it "Georgium Sidus" after King George III. Other astronomers called it "Herschel" after the discoverer. The astronomer Johann Bode suggested that it would be more appropriate to use the mythological name Uranus, which would match with the five planets that were named in antiquity. Despite the suggestion, the name Uranus was not commonly used until 1850.
The existence of the planet Neptune was predicted by two astronomers (John Couch Adams and Urbain Jean Joseph Leverrier), and when it was discovered with telescopes there was a debate about who should be allowed to name it. Leverrier actually wanted to name it after himself. However, the name Neptune was proposed and became the standard used by scientists.
Pluto was discovered in 1930 by Clyde Tombaugh at Lowell Observatory in Flagstaff, Arizona. According to the Nine Planets Website, other names suggested for Pluto included Lowell, Atlas, Artemis, Perseus, Vulan, Tantalus, Idana, Cronus, Zymal and Minerva (suggested by the New York Times). The name Pluto was apparently suggested by Venetia Burney, an 11-year-old from Oxford, England, and then recommended to astronomers by the observatory staff. Pluto won out, possibly because it's appropriate for the most distant world to be named after the god of the underworld.
Pluto's moon was named by its discoverer, James Christy, who found the moon in 1978 when studying photographic plates of Pluto. Apparently he wanted to name it after his wife, Charlene, but the nomenclature rules in astronomy wouldn't allow this. However, when he was looking for a different name he came across the Greek mythological figure Charon, which included the first part of his wife's name. Plus it was very appropriate since Charon was the ferryman who carried people to the underworld, which fits very well with the name of its planet, Pluto!