Direction of the force on eddy currents  
When a current passes through the coil, I am happy that if the metal disc rotates and cuts the flux, then eddy currents will be induced in the metal disc to oppose field of the electromagnet. 
My question is, what is the direction of the force on the currents, or how you work it out, as I know that the disc will stop as the force created will oppose that of the motion of the disc, but how can this be if the currents are on the surface of the disc?
 A: You're right that the magnetic force with oppose the rotation disc so the force on the eddy currents would be in the opposite direction of the disc's motion.
A simple way to think of it is to think of an eddy current circulating in the part of the disc that is about to pass between the poles of the magnet.  If the near pole is is a north pole then the eddy current would flow counterclockwise producing a north pole on the near side of the disc that would be repelled by the magnet's north pole.  At the same time, on the part of the disc that has just passed through the magnet and is moving away from it, there would be an eddy current current flowing clockwise which would produce a south pole on the near side of the disc which would be attracted back to the magnet.
A: If the electromagnet is producing a north pole as shown in the digram the magnetic field due to the electromagnet is going into the disc and is weaker the further away the disc is from the poles of the electromagnet.  

I do not know the exact shape of the eddy current loop but I do know that the direction of the eddy (induced) current will be such as to produce a south pole as shown in the diagram.
This is Lenz's law in action as that induced current is trying to oppose the motion producing it which is the rotation of the disc.
You can think of it as the induced south pole on the disc attracting the north pole of the electromagnet and so trying to reduce the speed of the disc moving away from the electromagnet.
There is an equivalent north pole induced on the left hand side of the electromagnet which is repelling the north pole of the electromagnet , again trying to reduce the rotational speed of the disc.
On the other side of the disc there are induced currents interacting with the south pole of the electromagnet again trying to reduce the rotational speed of the disc.

There are equivalent forces on the electron magnet trying to move it in the same direction as that of the disc.  
This video of Arago's disc is one of many video available on the Internet to show this effect.
