A "reversal" is the ability of a character to go straight from a non-hittable frame (such as being knocked down), directly into an special attack, with no delay or events in between. As an example, if Ken is knocked onto the ground, the player may enter the command for a shoryuken as he is getting up, which Ken will then execute in the very first frame after getting up from the ground.
There are four situations where a character may perform a reversal attack. The first is after getting up from being knocked down. The second is directly after exiting a blockstun state. The third is directly following your character landing from being hit in the air with a non-knockdown move. (The first frame after landing will be the special.) The fourth is directly after exiting a stun state. (Although technically a stun state isn't a non-hittable frame.)
Yay Google. The obvious benefit of Reversals is that it doesn't give your opponent much chance to hit you before your move comes out. Kindly, modern games like Street Fighter IV offer some leeway on the input of the special, so even if you do it a bit early, it still comes out as a Reversal.
In SFIV, any move that gets Reversal to flash up on the screen will, as an added benefit, break through the armour of a Focus Attack or any of the special moves that happen to have the same one-hit armour as a Focus Attack.