Saturn is about ten times as far from the Sun as we are - that is, close to a billion miles away. It takes thirty years to go around the Sun. Its axis is tilted, like Earth's is, which is why we can see its rings. It's about nine times Earth's diameter - about seventy thousand miles across.
Galileo was the first to see Saturn's rings, but didn't know what they were. Christiaan Huygens figured out what the rings were, and also discovered Titan, Saturn's largest moon. Giovanni Cassini discovered four more of Saturn's moons, and was the first to see that the rings had a gap in them (the Cassini Division). All three of them did their work in the 17th century.
Better telescopes showed us four more moons over the next 250 years, but until 1977 they were points of light in telescopes - we knew their rough sizes and masses, and of course we could watch their distances from Saturn and how long they took to orbit it, but that was about it... until we actually sent something out there to start taking pictures. We sent three flyby missions during 1978-1981 - Pioneer 11, Voyagers 1 and 2. Then, in 2004, the Cassini orbiter, carrying the Huygens probe, reached Saturn, and it's still there, eight years later. (So when I talk about what "Cassini" or "Huygens" found out, it's about the 21st-century probes, not the 17th-century astronomers.)
Some of the moons that I'm going to talk about, but not all, are "world-sized" - that is, big enough for its own gravity to pull it into a sphere. Your typical Saturnian moon orbits the planet's equator in a circular (non-elliptical) path, and keeps one face toward the planet all the time (the way Earth's moon does toward Earth). Most of Saturn's moons have no seismic activity, magnetic field, or atmosphere.
That's the typical moon. There are exceptions to every one of those rules - that's why this is so interesting to me.
The world-sized moons of Saturn, in order of distance from the planet, are: Mimas, Enceladus, Tethys, Dione, Rhea, Titan, and Iapetus. Except for Titan and Rhea being larger than Iapetus, that's also the order from smallest to largest. Two other moons are "borderline" whether they should get the same status: Hyperion (orbiting just outside of Titan) and Phoebe (extremely distant).
tavella mentioned that the number of Saturn's known moons has gone from about 20 last time she looked to something like 60 now. Yeah. My reaction is this. Saturn has, at least, billions of "moons" - meaning things orbiting it. The rings are made of "moons" - a mile or less in diameter, with no real division between "moons" and "ring material", except for "things big enough for Cassini's telescopes to discern them individually". It also has an undetermined but huge number of progressively smaller rocks orbiting it at extremely large distances - captured asteroids and centaurs and comets, going from barely-noticeable down to who-cares; each time telescope technology gets better we find a bunch more rocks.
They're rocks. Yeah, when we find them we give them names. But I'm not about memorizing names, not now that we know a lot more about things than their names. There are lots and lots of rocks orbiting Saturn. I'm mostly interested in the worlds - and only interested in a rock if it shows us something as interesting as a whole world.
So anyway, that's the textbook stuff. Boring part over.