A solid copper plate attached to the end of a pendulum swings between the poles of an electromagnet. With the electromagnet on, when the plate swings through the magnetic field, eddy currents generated in the plate exert a damping force on it. You can vary the damping by changing the current flowing through the electromagnet. With the current at maximum (10 A; 5 A per magnet coil), the pendulum slows dramatically at the bottom of its swing, and if it hasn't stopped dead, it does so on the backswing. When you substitute a copper plate with slits cut in it (resting on the table in front of the frame, in the photograph), there is essentially no damping, because the slits disrupt the eddy currents, and the pendulum swings freely.

The apparatus in the photograph is a pendulum, whose bob is a rectangular copper plate. At the bottom of its swing, the plate sits between the poles pieces of an electromagnet, which is fed by the power supply sitting on the middle shelf of the cart. With the power off, there is no magnetic field, and when you raise the copper plate and then release it, it swings freely between the pole pieces. When you turn on the power supply, there is a magnetic field between the pole pieces of the magnet. With the magnet connected as it is in the apparatus, the north pole happens to be on the left, and the south pole is on right, which means that the field points from right to left. If you raise the pendulum bob behind the apparatus, as it swings toward the front, the magnetic field induces an EMF that causes (positive) charges to flow downward. (Relative to the magnetic field, the charges in the copper plate are moving toward the front of the cart, and with B pointing to the left, they experience a force, F = qv × B, which points down.) The regions in front of and behind the magnetic field provide return paths for the downward-flowing charges, and this produces two sets of currents, one in front of, and one behind the pole pieces. If we view the apparatus from the left, the currents in front of the magnet flow counterclockwise, and those behind the magnet flow clockwise. Such currents, induced in an extended object (as opposed to a ring or a loop of wire) are called eddy currents.

As the pendulum bob swings forward, the charges flowing downward in the magnetic field experience a force, F = qv × B, which points toward the rear of the apparatus, in the opposite direction to the motion of the pendulum bob. The other parts of the eddy currents, which are outside the field, do not experience such a force, and there is a net backward force on the pendulum bob, which greatly slows it or stops it. Similarly, if you raise the pendulum from the front (or if it happens to pass through the bottom of its swing and be returning on its backswing), as the bob passes between the magnet pole pieces, the magnetic field induces an EMF that causes (positive) charges to flow upward. (v is now reversed, so qv × B points upward.) This produces two sets of eddy currents that flow in opposite directions to those produced when the pendulum swings forward. The charges flowing upward in the magnetic field now experience a force, F = qv × B, which points toward the front of the apparatus, again in the opposite direction to the motion of the pendulum bob. This, again, greatly slows or stops the pendulum bob. A device such as this, in which eddy currents slow a moving part, is known as an eddy current brake. In connection with oscillating systems, this phenomenon is also known as eddy current damping or magnetic damping. (See demonstration 40.39 -- Damping of eddy current pendulum.) It is perhaps most commonly used in mechanical balances, in which an aluminum vane attached to the end of the beam rides within a slot in the vertical member that holds the zero indicator. Magnets on either side of the slot induce eddy currents in the vane as the beam oscillates up and down, which slows the oscillation and reduces its amplitude so that the beam comes to rest more quickly than it otherwise would.

If the magnetic field in the pendulum apparatus pointed in the opposite direction, the motions of the currents in the middle of the field would be in opposite directions to those described above (as would the associated sets of eddy currents), but since the field now points from left to right, qv × B would still oppose the motion of the pendulum bob. In either case, if you substitute the slotted plate for the solid plate, the pendulum swings freely, with essentially no damping as the bob swings through the magnetic field. The reason for this is that the slots in the plate break the paths that the eddy currents would take, and the eddy currents cannot flow. In transformers, which have an iron core to improve the coupling between the primary and the secondary, the changing magnetic flux that induces an EMF in the secondary to produce current in it, would also induce EMFs in the iron core and produce eddy currents in it. This would cause excess heating and loss of efficiency. To minimize these eddy currents, transformer manufacturers use a laminated core. Instead of making the core out of solid pieces of iron, they stack thin layers of iron together. The layers have an insulating coating, so that current cannot flow from layer to layer. In the same way that cutting slots in the pendulum disrupts the eddy currents that would flow in a solid plate and virtually eliminates the damping, making the transformer core in layers disrupts the eddy currents that would flow in a solid core, and thus greatly reduces their effect.

References:

1) Sears, Francis Weston and Zemansky, Mark W. College Physics, Third Edition (Reading, Massachusetts: Addison-Wesley Publishing Company, 1960) p. 672-4.
2) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition (New York: John Wiley and Sons, 1977), pp. 788.