A wire carrying a current is surrounded by its own magnetic field. If the wire is placed in the magnetic field of a magnet, the field of the wire interacts with the field of the magnet. The result is a force on the wire. The magnet is pushed in the opposite direction with an equal force, so the magnet and the conductor exert a force on each other.
In the jumping wire demonstration, a loose wire lies across the magnetic field between the poles of a magnet. When the current is switched on, the wire jumps. The force is greatest when the wire is at right angles to the field. If the wire is parallel to the field, there is no force on it.
The direction of the force is reversed if the current is reversed, or if the magnet is turned round so that the field is reversed. The force is bigger with a larger current or a stronger magnet.