Proton torpedos are a weapon that is useful for targets that are too strong for other weapons. A single torpedo can destroy targets that might require multiple missiles or in some case, can't be destroyed by missiles. Only larger, shielded targets can be targeted by torpedos. These can often be identified by a pulse effect around the target. Since the torpedos take a few seconds to lock on to the target, they are only suitable for targets that don't move such as fixed points on capital ships.
Torpedo launchers are only available at level 50. Heroic space missions are the only missions where torpedos are essential, since some heroic missions have main objectives that can't be destroyed without them (and might even require more than one torpedo hit). Earlier space missions don't need torpedos for primary objectives, but often have bonus objectives that require torpedos, such as the bridge of a capital ship. Since these missions are usually done before level 50, those bonus objectives can't be completed unless you go back and do them again after reaching level 50. Some missions also have objectives that can be destroyed using a torpedo, but may also be destroyed using multiple missiles - such as the shield generators in the Clouds of Vondoru and Zosha Advance missions.
Most of the heroic space missions introduced in patch 1.6 have objectives that can't be completed without a torpedo launcher. In these missions objectives require 2 torpedo hits to destroy.
Your ship will only carry 4 torpedos per mission and the number is indicated next to the missile indicator as shown to the right. Firing a torpedo is similar to firing a missile since both are done using the right mouse button. Usually when you have a target, the right mouse button will fire a missile. When a torpedo suitable objective is lined up, your crosshairs will change to a different shape indicating you will fire a torpedo instead of a missile. Hold down your right mouse for a few of seconds until the torpedo locks on target and then release your button to fire. It is not always easy to keep the crosshair on a target long enough to get a lock.