Quick is a status effect of sorts that gives you the next active turn. Quick status can be gained by having the spell 'Quick' successfully cast on you or by having the reaction ability 'Critical Quick' triggered. The triggering of Critical Quick occurs when the unit with the ability equipped receives damage that leaves them with under 20% of their max HP.
While Critical Quick might not seem that great there are situations when it is the best reaction ability in the game. For a more reliant way to gain Quick, learn it for 800 JP from the Time Mage skill set. Quick costs 24 MP which might be a little costly but has decent speed (25) and a decent formula (targetsfaith*castersfaith*(MA+140)). Rig some zodiac compatibility between your group and Quick can be very reliable and fun to use.
Quickk can be reflected (though not Math Skilled) but the most interesting thing about it is that it can be Counter Magicked! Seriously, that exclamation point isn't one of the fake ones that should be a period, I actually used to get giddy when considering this. Look at the spells that can be Counter Magicked:
All Black Magic except Fire 4, Ice 4, and Bolt 4
Holy, Dark Holy
Yin-Yang Magic except Pray Faith and Doubt Faith
Slow, Slow 2, Don't Move, Quick, Stop, Demi, Demi 2, Meteor
Bio, Bio 2, Bio 3
Melt, Tornado, Quake
Attacks from the Blaze Gun, Glacier Gun, and Blast Gun
So just how does Quick fit into all this? With the proper setup you can orchestrate a near-endless chain of Quicks within your team (sometimes referred to as The Quickening), acting and acting while almost no in-game time passes and the enemy sits there staring at your chain of constant action.
Ramza: "Restore lost time in a spiral line! Quick!"