Ex Special Moves are initiated by pressing two of the attack buttons instead of one while initiating a special move. An example of this would be an EX hadouken being a quarter-circle toward the enemy and a light+medium punch. It requires one meter of super bar to initiate one of these moves on the fly. Personally, I found using a best of 7 with 99 seconds set up to be the best.
Personally, I paired these with my Pandora achievements given the fact once you initiate Pandora you instantly have a full super meter. This way, you can execute 3 of these moves per Pandora and knock out the trophy in no time. Don't forget, they have to connect to count. It's because of this that local versus is your best bet. Second controllers are preferred however it's still useful to fight AI in this mode given the fact you can adjust the health bar. Give yourself and the AI/Second controller 25% handicaps and initiate Pandora at the start. Then fire your 3 Ex Special Moves and win and you'll be making progress towards both achievements!
Ex Special Moves can also be initiated when you use a characters special move that can be charged in to their Super Move. When your character flashes after a couple seconds, release the attack button you had held and you'll launch an Ex Super Move. An example of this is holding Ryu's Hadouken by pressing Quarter Circle Toward and then any punch. Hold the punch, and he'll start charging the Hadouken. If you release it after the flash, It's an Ex Super Move and if you hold it all the way through it's a Super Art.