The local coordinate system is the coordinate system that relates specifically to the selected object.
A book in object space rests on a table in world space. The book has its own local coordinate system.
Each object has its own local center and coordinate system as defined by the location and orientation of the object's pivot point. The local center and coordinate system of an object combine to define its object space.
The direction of the object's X, Y, and Z axes depend on the current transforms of the object. Contrast with the world coordinate system.
You can see the difference between the two coordinate systems when you unintentionally rotate an object, such as a wheel on a car model, around the world axis instead of the object's local axis. The wheel immediately flies off in a large arc because the center of the rotation is at the origin of the world coordinates.
To rotate the wheel correctly, first change the coordinate system to Local, using the pop-up list on the toolbar. The wheel then rotates around its own hub, which is the origin of its local coordinates.