The fight pits are are special mission you can run any time after conquering the fortress in a region- the quest marker will appear by the local arena. This first marker will appear as 'Fight Pit - Maggots,' indicating that these are the weakest fighters of this particular pit. The pit fight itself is exactly the same as the fights assigned from the army screen (i.e. you can't interfere in any way), but if you win you unlock a new quest - 'Fight Pit- Warriors.' If a follower of yours succeeds here, then the final iteration is unlocked- 'Fight Pit - Champions.' Beating this final round unlocks the last defender slot for your fortress (allowing you to field more captains when resisting a siege). Beating the fight pits in Minas Morgul unlocks a special skill, and beating all of them unlocks an achievement.
The fight pits are a great place to get replacements for low level captains in the endgame- send your level 12 guy in, then Dominate the level 40 fighter that annihilates him. Every so often you'll encounter some Iron-Willed s.o.b. that breaks the cycle, but it's still quicker than hunting down captains in the field.
Do note that an enemy captain who wins his match levels up immediately- if he was exactly your level when the fight started, he'll be too powerful to Dominate once it's over.
Tactically, you of course want to match strength to weakness- sending a fire-weapon'd captains after pyrophobic prey, and so forth. Orcs with elemental weapons in general seem to have in edge in the arena, as do orcs with gangs of ologs, or savages, or stun bombs,