Special Moves by default are done by inputting a command motion, then pressing a certain button. If you turned on Release Check, you also have the option to perform your special move by holding down the button, performing the motion, then releasing the held button. Most of them serve as a combo ender after your basic combo string.
As stated earlier, turning on Alternate Control in the Controller Preset menu changes the inputs for special moves into Street Fighter-type controls, as opposed to the default Mortal Kombat-type controls.
By pressing the Meter Burn button after the input for a special, you will be able to perform the Meter Burn version of that special move, which is very similiar to the EX moves of most fighting games. This expends one bar of the super meter. Certain properties of special moves also change when they are done using Meter Burn, such as added tracking, added armor, added projectiles, etc. Keep in mind that not all special moves have Meter Burn versions, you can check the in-game movelist to view which ones can be powered up or not.