I have tried pressing all three of my mouse buttons when playing game on PC.. The left button fires lasers, The middle button fires the missiles and the right button fires nothing. I am trying to figure out how to use the plasma cannon. I don't see it in my ship equipment , so I am not sure that I even have it.

The term "Plasma Cannon" is bad as it should have been labeled "Plasma Ability" in the game's controls. After your first upgrade for your ship you'll be able to get an ability that boosts your weapons. This ability can be gained via your logbook. Once you have gain this ability your right mouse button should activate it, which will be indicated by a yellowish filter appearing over your HUD. This ability is the "Plasma Cannon" which is horribly named as it isn't a cannon.