The Galilean moons are the four largest of Jupiter's moons: Io, Europa, Ganymede, and Callisto. They were the first objects to be discovered by telescope, making them quite important in the history of astronomy. Galileo originally named these the Medicean stars1, although he quickly realized that they were actually moons. These moons are also sometimes referred to as the Galilean satellites.
Galileo Galilei had already made a big splash in theological and scientific circles with his extraordinary claims about the 1604 supernova, commonly called Kepler's Supernova. He noted that no parallax could be observed from the Earth's surface, and concluded that it was indeed a distant star, attacking the idea of an immutable heaven. In 1609 he constructed a simple working telescope, and in January of 1610 he documented the first stellar objects that were invisible to the unaided human eye.
This was an exciting scientific advancement, but it quickly became apparent that it was also a theologically important finding. On January 7, 1610 Galileo documented finding three new stars next to Jupiter; on the following nights he noted that they moved in ways that stars should not. On the 10th, one 'star' disappeared, to reappear a few nights later. (And, incidentally, on the 13th he found a fourth 'star' near Jupiter). Galileo made the connection that this could be caused by if they were orbiting Jupiter -- something that was not allowed in the strict geocentric view, which held that Earth was the center of rotation for the entire Universe.
Although Galileo's observations were immediately disputed by a large segment of the scientific community, his observations were confirmed by respected astronomer and firm geocentrist Christopher Clavius. While Clavius remained convinced that the geocentric model was basically correct, he was forced to accept that Jupiter did indeed appear to have moons. After this confirmation Galileo was widely regarded as a great scientist and even more well-respected by the scientific community than he had been previously. Unfortunately, he was also causing more trouble for the Catholic church with his observations of Venus (which pretty clearly did not orbit the Earth) and his defense of Copernicus; in this context Jupiter's moons were comparatively uncontroversial, but gave him considerable credence among the scientific community, making him 'dangerous'. As his attacks on the Ptolemaic system became more serious, the Church became more antagonistic, eventually leading to his imprisonment2.
Since that time we have discovered 63 additional, and much smaller, Jovian moons, but these were not known to Galileo, and are not considered Galilean moons. The Galilean moons are particularly impressive, being some of the largest non-planetary bodies in the solar system. Each of them are larger than Pluto or any of the other Dwarf planets, and Ganymede is larger than Mercury (although less massive).
The moons went through a long period during which they were called simply Jupiter I, Jupiter II, etc. But we currently we use the same names that were given to them by Simon Marius3 in 1614, a German astronomer who claimed to have discovered the moons just before Galileo4. In naming the moons he followed the suggestion of Johannes Kepler, and named them after the lovers of the Greek god Zeus (Jupiter being the Roman incarnation of Zeus): Io, Europa, Ganymede and Callisto.
These moons are so bright that were they not so near to Jupiter's glow, they would just barely be visible to the naked eye. They are still large and bright enough to be seen with a strong pair of binoculars or even the most basic telescope, and they remain a favorite target for beginning amateur astronomers. They are also all interesting bodies in their own right, and are well worth reading about in more detail in their own nodes.
1. He named these stars/moons after a former student and quite wealthy man, Cosimo II de' Medici, and at the time these bodies were known as the Medicea Sidera ('Medicean stars'), or alternatively, Cosmica Sidera ('Cosmo's Stars'). (Galileo also dedicated his book Sidereus Nuncius to Medici; these tributes shortly led to a patronage).
2. The church's action was fairly effective, politically speaking, and the following decades were full of attempts to construct working models that would support geo-heliocentrism -- models in which the Sun revolved around the Earth and the other planets revolved around the Sun.
3. In his book Mundus Jovialis ('The World of Jupiter'), 1614.
4. Galileo was not happy about this, and disputed the claim, but there is no firm evidence either way.