Basically, a reversing contactor is used to reverse the rotation of a motor. A non-reversing contactor just turns the motor on and off acts like a relay, it is just a single simple contactor, with three main contact terminals (1 per phase for a 3-phase supply to the motor.)

A reversing contactor is an assembly of two non-reversing contactors, with a mechanical and electrical interlock between them, so only one contactor can be closed (giving power supply to the motor) at any time. 

A reversing contactor interchanges two poles on a THREE PHASE (typically motor) application, such that the output are 120 degrees opposed from the input . This is useful in a conveyor, for example. A three phase motor that ran clockwise normally would run counterclockwise when controlled via reversing contactor.

The Forward/Reverse - Up/Down - Left-right etc. are usually opposite motions, and the opposite motions are obtained via labelled switching buttons which energizes the coil of the correct contactor, mechanically locking the other motion contactor and also electrically preventing it from closing.

In the standard 3-phase squirrel-cage rotor motor, the reverse motion is easily obtained by swapping any two of the 3 phase wires to the motor, which then runs in the opposite direction = reverse.

A non-reversing DC contactor is a SPST relay that makes a circuit when a coil is energized. When it makes contact, it is dedicated to only one direction of flow. A reversing DC contactor will make contact on electricity flowing in both directions when the coil is energized because it is DPDT. The contactors are completely different from one another.

A 3-phase standard non-reversing contactor wiring diagram is shown below, and has been called "Relay".