The underground corridors, as I'm sure everybody who took a tour of it (or saw Gladiator, for that matter) is well aware, is where the gladiators would prepare for the fight. Animals, slaves, and fighters were kept here and could be introduced into the middle of a fight by a series of lifts. Many say that this is the most interesting part of the Colosseum, showing off the Romans' architectural brilliance, as well as their flare for the dramatic.
Further, since these corridors were buried underground throughout much of history, they are much better preserved than the rest, which has been subject to thousands of years of the elements.
The third ring that has opened (previously you could only go to the first and second) would be where the lower class Romans (plebeians) sat for the fights (even if, I've always thought that they had it better than the rich patricians, who sat so close that they risked death by a stray arrow or javelin). However, aside from the better view of the inside, the third ring also offers great views of Rome itself.