When metal moves through a spatially varying magnetic field, or is located in a changing magnetic field, induced currents begin to circulate through the metal. These currents are called eddy currents because of their similarity to eddies in a flowing stream. In the case of the eddy current brake, a rotating disk has a magnetic field passing through it perpendicularly, but it is only strong in the area where the magnet is. The currents in that area experience a side thrust, which opposes the rotation of the disk. This interaction of field and current results in the "braking" of the disk, and thus the name "eddy current brake." The return currents close via parts of the disk where the field is weak, so there is a drag force only in the "generating" region.
It can be pointed out to advanced students that : (A) this effect is very hard to calculate for common magnetic field and disc geometries; (B) if the disc is slitted radially over the region covered by the magnet, the calculation becomes much easier, since the currents then must close through the rim and hub, whose resistance can be estimated; and (C) if the magnetic field is uniform everywhere in the disc, you have an open-circuited Faraday generator, and an electrostatic field builds up to cancel the v X b force, so no current flows, and there would be no braking effect.