The magnetic field of the disk magnets exerts a force on the electric current flowing in the wire. The wire will move up or down or forward or backward, depending on the direction of the current and the direction of the disks' magnetic field.
To predict the direction of movement, you can use a mathematical tool called the right-hand rule. Put your right hand near the section of wire that goes between the disk magnets. Make your hand flat, with your thumb sticking out to the side—your thumb should be at a right angle to your fingers. Place your hand so that your thumb points along the wire in the direction that the electric current is flowing (current flows from the positive terminal of the battery to the negative terminal) and so that your fingers point from the north pole of the disk magnets toward their south pole. (You can find the north pole of the magnets by using a compass; the south end of a compass will point toward the north pole of a magnet.) Your palm will then naturally "push" in the direction of the magnetic force on the wire.
The deflecting force that a magnet exerts on a current-carrying wire is the mechanism behind the operation of most electric motors. Curiously (and happily for our sense of symmetry!), the reverse effect is also true: Move a loop of wire across the pole of a magnet, and a current will begin to flow in the wire. This, of course, is the principle of the electric generator. The electric current you generate by moving this single loop of wire through the weak magnetic field of the disk magnets is too weak to detect with all but the most sensitive of microammeters.