On January 7, 1610, physicist and astronomer Galileo Galilei turned his new telescope to the nocturnal sky to watch the planet Jupiter and discovered the eponymous four moons of Jupiter, Ganimede, Callisto, Io, and Europa although he is not able to distinguish the last two until the following day.[1,2]
Based only on uncertain descriptions of the first practical telescope which the Dutch lens maker Hans Lippershey  tried to patent in the Netherlands in 1608, Galileo Galilei, in the following year was able to improve the telescope, with a magnifying capability of about 20 times. Moreover, Galileo was also one of the very first to try the telescope not only for terrestrial observations but also towards the nightly sky. With his improved telescope, he was able to see celestial bodies more distinctly than was ever possible before. This allowed Galilei to discover sometime between December 1609 and January 1610 what came to be known as the Galilean moons.
Three fixed stars, totally invisible by their smallness
On 7 January 1610 Galileo observed with his telescope what he described at the time as “three fixed stars, totally invisible by their smallness“, all close to Jupiter, and lying on a straight line through it. On the very same day he wrote a letter containing the first mention of Jupiter’s moons. At the time, he saw only three of them, and he believed them to be fixed stars near Jupiter. He continued to observe these celestial orbs from January 8 to March 2, 1610. In these observations, he discovered a fourth body, and also observed that the positions of these “stars” relative to Jupiter were changing in a way that would have been inexplicable, if they had really been fixed stars. Within a few days he concluded that they must be orbiting Jupiter. Galileo named the group of four the “Medicean stars”, in honour of his patron, Cosimo II de’ Medici and Cosimo’s three brothers. Later astronomers, however, renamed them Galilean satellites in honor of their discoverer.
Naming the Galilean Moons
The names that eventually prevailed were chosen by Simon Marius, who discovered the moons independently at the same time as Galileo : he named them at the suggestion of Johannes Kepler  after lovers of the god Zeus (the Greek equivalent of Jupiter): Io, Europa, Ganymede and Callisto, in his Mundus Jovialis, published in 1614. Galileo steadfastly refused to use Marius’ names and invented as a result the numbering scheme that is still used nowadays, in parallel with proper moon names. The numbers run from Jupiter outward, thus I, II, III and IV for Io, Europa, Ganymede, and Callisto respectively.
Importance of the Discovery
Galileo’s discovery proved the importance of the telescope as a tool for astronomers by showing that there were objects in space to be discovered that until then had remained unseen by the naked eye. More importantly, the discovery of celestial bodies orbiting something other than the Earth dealt a blow to the canonical Ptolemaic world system, which held that the Earth was at the center of the universe and all other celestial bodies revolved around it. In his writings Sidereus Nuncius (Starry Messenger), which announced his new celestial observations, Galileo does not explicitly mention the new Copernican heliocentrism, a theory that placed the Sun at the center of the universe. Nevertheless, Galileo accepted the Copernican theory.
The Longitude Problem
As a result of these discoveries, Galileo was able to develop a method of determining the current longitude based on the timing of the orbits of the Galilean moons. The times of the eclipses of the moons could be precisely calculated in advance, and compared with local observations on land or on ship to determine the local time and hence longitude. The main problem with the technique was that it was difficult to observe the Galilean moons through a telescope on a moving ship; a problem that Galileo tried to solve with the invention of the “celatone”, a piece of headgear with a telescope taking the place of an eyehole.
However, Galileo might not have been the first to get a glimpse of the moons of Jupiter. A Chinese historian of astronomy, Xi Zezong, has claimed that a “small reddish star” observed near Jupiter in 362 BC by Chinese astronomer Gan De may have been Ganymede, predating Galileo’s discovery by around two millennia.
Barth Netterfield, Astro 101 Class 17: The Jovian planets and their moons., 
References and Further Reading
-  The Galileo Affair, SciHi Blog, SciHi Blog, February 13, 2014.
-  Galileo Galilei and his Telescope, SciHi Blog, August 25, 2012.
-  Hans Lippershey and the Telescope, SciHi Blog, October 2, 1813.
-  Simon Marius and his Astronomical Discoveries, SciHi Blog, January 20, 2015.
-  The Galilean Moons at wikipedia
-  Debarbat, S. and C. Wilson: The Galilean Satellites of Jupiter from Galileo to Cassini, Roemer and Bradley in Planetary Astronomy from the Renaissance to the Rise of Astrophysics, Part A: Tycho Brahe to Newton; edited by R. Taton and C. Wilson in The General History of Astronomy vol. 2A, ed. by M. Hoskin p. 144–158. New York: Cambridge University Press 1989.
-  Extracting the Stopper, The Renaissance Mathematicus, June 2, 2010.
-  Well no, actually he didn’t. The Renaissance Mathematicus, April 13, 2016.
-  And Kepler Has His Own Opera – Kepler’s 3rd Planetary Law, SciHi Blog
-  The Galilean Moons at Wikidata
-  Barth Netterfield, Astro 101 Class 17: The Jovian planets and their moons., Barth Netterfield @ youtube
-  Timeline of the discovery of the moons of Jupiter, via Wikidata