Galileo was the first to see moons orbiting Jupiter. Why was this significant? In 1543, a lifetime earlier, Nicolaus Copernicus proposed his heliocentric system - the model postulating that the planets all orbit the sun (in opposition to the classical geocentric model which held that everything orbited the earth). Around 1605, Galileo's time, Johannes Kepler was modifying the Copernican model to ellipical orbits rather than circular ones.
Galileo's observation was a strong piece of evidence against a strict geocentric model of the solar system - here were objects orbiting something other than earth. This use of the telescope by Galileo was the beginning of the end for the geocentric model. Unfortunately, the Catholic church at the time held to the geocentric model as being the one in accord with the Bible. And the Catholic Church in early 1600s Italy was not to be mocked. By the way, Martin Luther, leader of the Protestant Reformation, also disbelieved the heliocentric model of Copernicus so it wasn't just the Catholics.
As an aside, there are lots of similarities here between 17th century geocentrists and modern-day evolution deniers. Both claim that a clear, literal reading of scripture supports their position.
The I, E, G, and C stand for Io, Europa, Ganymede, and Callisto - the four Galilean moons of Jupiter from closest to furthest from the planet. Tonight you would see Io on the left and Ganymede and Callisto on the right (Europa is in Jupiter's shadow). In a couple of days (January 12), the view would be:
Io still on the right with Europa (now visible), Ganymede, and Callisto (must closer now) on the right. Every night will show a different view.
Here's what it looks like through a telescope (from Wikimedia Commons).
A never-ending dance of light as the moons orbit Jupiter. In 1676, Danish astronomy Ole Romer even calculated the speed of light from the movements of Jupiter's moons (maybe I'll write about that sometime - it's a cool story). I also have my students in astronomy class do an exercise (Project CLEA) where they calculate the mass of Jupiter from the orbital periods of the moons. It's amazing what you can learn by just observing (and a little thinking). Too bad there are still people opposed to that (see the above comment about creationists).