One of the most iconic pictures from Voyager 1, as it flew through the Saturn system in 1980, was this picture of Mimas. Its resemblance to the Death Star, just after Empire Strikes Back was released, was a cool and almost spooky coincidence. It looks like it does because of the huge crater (Herschel) taking up a third of one side of the moon. That crater is interesting; Mimas isn't the only one of Saturn's moons to show clear evidence that large amounts of debris was flying around the system at some point in the very distant past. The crater is so big that some models say it should have shattered Mimas when it hit, so why it didn't is a good question.
It's not the only thing interesting about Mimas, though. The moon's gravitational effects had been known for two hundred years before it itself was discovered: when Cassini (the seventeenth-century astronomer, not the space probe) realized that the strange shape of Saturn was a ring around a sphere - which Galileo (also the seventeenth-century astronomer, not the space probe) had seen but didn't realize what he was looking at (Galileo called them "handles") - he (Cassini) also saw that the ring was not a single thing, but had a gap in it.
The gap is known today as the Cassini Division, and it exists because of Mimas: a particle that tried to orbit in the Division would orbit Saturn in exactly half the time Mimas takes to orbit; this means every other time around Saturn it would get a pull from Mimas, always in the same direction, and it would either lift into a higher orbit or drop into a lower one.
Mimas is also resonant with Pandora, which I talked about the day before yesterday - Mimas goes around twice for every three orbits of Pandora (on average; remember that the orbit varies just a bit because Prometheus gives it a kick every so often). And there's also resonance with Tethys - but I'll get to that shortly.
One other thing about Mimas that we'd like to know someday. We know of at least four moons of gas giants that are geologically active: Io and Europa around Jupiter, Enceladus (I'll get to that one, I promise) around Saturn, and Triton around Neptune. In all four cases, we're pretty sure that the reason they're geologically active is that, orbiting as close as they do to a huge body, they're subject to tidal forces that flex the moon and produce molten cores, which result in eruptions (of lava or water, depending on what the moon is made of). But not Mimas. It's the closest world-sized, sphere-shaped moon of Saturn, it's barely any smaller than Enceladus, it's clearly subject to stronger tides than Enceladus, and it's made of the same thing (mostly ice). So why doesn't it flex and melt its ice, like Enceladus does?