That game would be Mario and Luigi Superstar Saga..

Of course, it's probably easier to think of it in Mario RPG and Paper Mario terms, but instead of simply reducing damage, you can avoid it entirely or reflect it back. If a goomba attempts to attack you by running into you, you can jump over it normally - but if you time it correctly, instead of jumping over the attack, you can land on the goomba directly, and deal some damage. Of course, if you screw up you'll be hurt.

Of course, they might attack Mario or Luigi, so you either have to hit the A button or the B button at the correct time based on who's attacking who. If you hit A instead of B, Mario will attempt to avoid an attack that's actually attacking Luigi. And a lot of them really mess with your head - there's this one enemy that'll flash green when it's attacking Luigi and red when it's attacking Mario. This gets you used to the whole color association thing. Then, if you hurt it enough, it screws up and attacks the opposite color that it flashes, and pretty much lures you to avoid with the wrong person. It's hard to explain how much it affects you unless you've played it.