Of course, Galileo isn't the only astronomer who claimed to have discovered the Galilean Satellites. Simon Marius, a German astronomer, claimed at the time to have observed them in December 1609. Because Galileo published first, Galileo is now generally regarded as the discoverer, but Marius' names for the Galilean satellites have stuck: Io, Europa, Ganymede, and Callisto.
A good overview of Galileo's observations, published in Sidereus Nuncius in March 1610, can be found on the JPL website, but here is the translation of the pertinent part:
On the 7th day of January in the present year, 1610, in the first hour of the following night, when I was viewing the constellations of the heavons through a telescope, the planet Jupiter presented itself to my view, and as I had prepared for myself a very excellent instrument, I noticed a circumstance which I had never been able to notice before, namely that three little stars, small but very bright, were near the planet; and although I believed them to belong to a number of the fixed stars, yet they made me somewhat wonder, because they seemed to be arranged exactly in a straight line, parallel to the ecliptic, and to be brighter than the rest of the stars, equal to them in magnitude . . .When on January 8th, led by some fatality, I turned again to look at the same part of the heavens, I found a very different state of things, for there were three little stars all west of Jupiter, and nearer together than on the previous night.Don't forget, only one more year until the 400th anniversary of Io's discovery.
Link: Discovery of the Galilean Satellites [www2.jpl.nasa.gov]