If a large conductive metal plate is moved through a magnetic field which intersects perpendicularly to the sheet, the magnetic field will induce small "rings" of current which will actually create internal magnetic fields opposing the change. This is why a large sheet of metal swung through a strong magnetic field will stop as it starts to move through the field. All of its kinetic energy will cause a major change in the magnetic field as it enters it which will induce rings of current which will oppose the surrounding magnetic field and slow the object down. In effect, the kinetic energy will go into driving small currents inside the metal which will give off that energy as heat as they push through the metal.

If this isn't a satisfying answer, consider a simple wire loop being moved through a magnetic field. If you've learned anything about motors and/or generators, you will have probably learned that a current will be induced in this loop in a similar fashion. Likewise, a wire loop being pushed into a magnetic field will induce a current which will make it difficult to continue pushing. Likewise, it will resist being pulled out as well. An eddy current does the same thing, but instead of being forced in the path of the loop, it is allowed to travel in the "eddy" pattern that nature provides.

To get rid of eddy currents, slits can be cut in metals so that large eddies cannot occur. This is why the metal cores of transformers are often assembled in small laminations with an insulator in between. This prevents AC energy from being lost to eddies generated within the magnetic core (which typically is also conductive because it is a metal like iron).

Now, sometimes eddy currents are a good thing. Mentioned above, eddy currents help turn kinetic energy quickly into other forms of energy. Because of this, braking systems have been created which take advantage of it. Adding a magnetic field around a spinning piece of metal will cause eddy currents in that metal to create magnetic fields which will slow the object spinning down quickly as long as the magnetic is strong enough.