EX Special Moves are slightly stronger versions of other special moves that use up some of your Special Gauge, they are activated by pushing 2 Kicks or Punch buttons at the same time when you use the move. For example Ryu can use Hadoken, Quarter Circle Forward + Any Punch, or EX Hadoken, Quarter Circle Forward + Any 2 Punches.
However there is an alternate way to activate EX moves in this game, when you use the move continue holding the button and you will charge it, when your character flashes red for a second you can release the attack as an EX Version without using any of your Special Gauge.
If you want to farm these the best way I have found, so far, is to use Ken in arcade mode. Begin the battle by using Quarter Circle Back + Heavy Kick, as soon as that move ends use Heavy Shoryuken, this will knock the opponent down. Quickly do Quarter Circle Back and then hold the kick, you should be able to charge it to an EX before they get back up. Use it on them and then immediately use Heavy Shoryuken to knock them over and keep repeating this process until you win. You also can use some EX Shoryukens in place of the heavies since you will be gaining meter from your attacks.
I assume EX Moves in Local Versus count so that way is probably easier then what I said, but it is only an assumption, not a fact.