I wouldn't call it a mini game...
basically there is a status bar in the top right and about 1/3rd of the way in it has a white line with a grey section trailing off to the right.
when you tap the reload button a 2nd white bar travels across the status bar from left to right. If you don't do anything your character will spend that entire time reloading the gun, (the speed varies depending on the weapon and it seems to slow down slightly over the white/gray section) This can take as long at 5 seconds to reload for some of the slower guns like the sniper rifle (which BTW only allows one shot between reloads).
so it looks like this:
[.......I||||........................]
where the I is the white line and the ||||is the gray area
You can speed this up by tapping the reload button a 2nd time when the 2nd white bar is over the white/gray area, your character will slam the ammo into the gun perfectly and you can carry on with a really fast reload.
In addition to that if you tap it when it's over the small white bar before the gray area your ammo will get a boost in the damage it can cause, which can be beneficial to taking out your enemies faster.
Unfortunately if you hit it at the wrong time. too early or too late, your character will fumble the ammo and the gun will jam making it take up to twice as long to reload.
This works really well online because it forces you to actually think about the reloading process instead of just blindly pounding that button or spraying your opponents with ammo and letting the reload procedure take over automatically.
once you get good at it, it the timing for a perfect reload will become somewhat of a second nature, if you're not in battle it's pretty easy to hit at least the gray area every time but I've been caught with my pants down more then once where I was being chased down, got too anxious when I hit the reload button, and screwed myself.