Short answers, pulled from the story.
Galileo first observed the moons on the 7th of January 1610 and recognized them as satellites orbiting Jupiter by the 15th of January 1610. He published his findings in the Sidereus Nuncius in March 1610.
Simon Marius gave the moons their present names in his Mundus Jovialis, published in 1614. He named them after mythological figures associated with Zeus, on the suggestion of Johannes Kepler, whom he met at the Ratisbon fair in October 1613.
Ganymede is the largest moon in the Solar System at 5,262.4 kilometers in diameter, surpassing the planet Mercury in size. Callisto is barely smaller than Mercury, while Io and Europa are roughly the size of Earth's Moon.
Europa is considered the strongest candidate for extraterrestrial life. It has a layer of water thought to be 100 kilometers thick beneath its ice crust, kept liquid at the bottom by tidal heating from Jupiter's gravity. Callisto also has a possible subsurface ocean, but this is considered less likely to harbor life than Europa.
The eclipses of the moons by Jupiter could be calculated in advance with precision, and comparing those predicted times with local observation times allowed navigators to determine longitude. Galileo applied for the Spanish longitude prize of 6,000 gold ducats in 1616 and later applied for the Dutch prize as well.
Io, Europa, and Ganymede are locked in a 4:2:1 orbital resonance: for every orbit Ganymede completes, Europa completes two and Io completes four. Callisto is outside this resonance but is expected to be captured into it in roughly 1.5 billion years, creating a 1:2:4:8 chain.