During matches in Arcade Mode, a prompt will appear below your character's health bar. Complete the command to get a boost to your super bar and more points (this helps unlock characters/endings in Arcade mode).
Here are the ones I've seen:
Jump 3 times.
Just like it says - do three jumps. Hops and super jumps count, but getting launched into the air by an enemy's attack does not.
Dash/roll 3 times
Like above, roll forward or backward three times to complete.
Hit opponent with a special move
Any moves included in the special move list of your character will work, and it even counts the hit if your opponent blocks it.
Hit opponent with an EX move
Same as above, the hit can be blocked and still count.
Hit opponent with 5 normal attacks
Same as above, the hits can be blocked and still count.
Hit opponent with a 3 hit combo
This combo does need to hit - it cannot be blocked. Most characters can jump in with a deep hit and continue the combo with two ducking light kicks.
Hit opponent with a desperation move
Any super move will do.
This is the only one that isn't self explanatory - if you play the tutorial all the way through (and there's an achievement for it, so why not?), you'll get an easy example with Kyo. You need to hit with a special move, then cancel into another.
These do not need to be done in the same fight, or even the same playthrough of Arcade mode. Just keep working at it, and you'll get it done. If you concentrate on just doing these target moves, you can usually get 5-6 done per fight.
They will only show up in regular fights in Arcade mode. You cannot do target actions during the "Here Comes a New Challenger!" matches or the boss fights.