Ah, Quickenings... They're possibly my favorite part of XII's battle system.
Here's a good overview of them, but I'll try to go more in detail as to how to use them. I'm trying to be as detailed as possible, so if I'm saying stuff you already know, don't be offended, OK?
sweatdrop You have all of them, meaning each character has all three of theirs? That means your MP gauge is split in three, and when each part is full, it turns orange, right? That's how many Mist Charges you have, and you can use the corresponding Quickening. For example, if you have one Mist Charge (the gauge is 1/3 full, with one orange section), you can use just a level one Quickening. If your MP is full, though, you can use any of the three.
So to select your Quickening, go to Mist > Quickenings > Whichever one you want to or can use, and voilĂ ! Whatever Mist Charges you needed to use that Quickening will be used up. For example, you had three and used a level one Quickening, so you have two left.
Then we chain! On the bottom right are your characters, and you can use their Quickenings before your first character's ends, based on the Mist Charges they have. It works the same way as the first time, with the level of Quickenings you can use and the number of Mist Charges. This time, though, you can't choose the Quickenings you use. It's randomized, and if you don't have enough Mist Charges to use the Quickening that comes up for a certain character, it's greyed out. Otherwise, you can then activate their Quickenings with triangle, square, or cross. That Quickenings will then follow the current one, and the process repeats.
The other option that will randomly appear is Mist Recharge. This will fill that character's Mist Charges back up to three, but once the Quickenings end, they won't regain the MP. It does, however, let you chain more Quickenings with that character.
The last option you have is pressing R2, which will reshuffle the options you're given. If you don't have any usable options (if you don't have enough Mist Charges for any of the Quickenings shown), R2 will give you another shot and give you new options. They'll also be shuffled when you select a Quickening when you have the next opportunity to chain another.
The last part, as you may know, is the timer. You only have a few seconds at most to choose the next Quickening to chain, or it will just end and break your chain. Every time you chain, you get less time for the next one. Mist Recharges are reshuffling both take up a significant amount of time, so avoid using them if you can once you start running out of time.
Oh, wait, I lied. The last part is the concurrences. The page I gave explains them pretty well; depending on how many of which level Quickenings you use, you can get stronger concurrences, which are one last attack after you chain your last Quickening.