The action of a firearm is a descriptive term for the kind of mechanism it has.
Bolt action, for example, refers to an action that operates like an ordinary door bolt, where a handle is moved to unlock the mechanism and then the bolt is slid backwards and forwards to unload and reload the firearm.
There are lever-action firearms, where a lever, typically at the bottom of firearm, is pushed down to open the mechanism and then raised to close it again.
The semi-automatic action is one where much of the loading action is done automatically by the firearm. It's kind of like having an automatic transmission on a car versus a standard. The same operations occur, but the firearm does it automatically for the user.
There are many types of actions. Those are some common ones.