What are the differences between direct law and mechanical back-up?
The key piece of knowledge to understand a FBW system is that you need to differentiate between signalling and actuation.
signalling is what tells the control surface to move and to what position. In old airplanes, this was by steel wires. In FBW airplanes it's by copper wires carrying an electrical signal.
actuation is what actually moves the controls surfaces to the commanded position, the muscles. Traditionally this is hydraulics, but you can also find electric actuators these days.
For the Airbus, Normal, Alternate &amp; Direct all use electrical signalling (whether produced by computers or not). But what to do if there's no electrics? That's Mechanical backup, where you have steel-wire signalling of specific surfaces.