LIMs: When a piece of metal passes through a magnetic field, a current is induced. This principle is used all the time in your home with transformers (the big, box outlet plug-ins), but those are rotary induction motors since they spin. What those do is change AC current to DC current. When the power supplied to the magnets in the transformer turns on and off quickly (AC current) then the metal coil in between the two batteries spin. Now if you take that and lay it out flat, you get a linear induction motor. Two magnets are placed on top of each other on the side of the track, leaving a space in between. When a metal fin attached to the train passes through the magnetic field created, it "rides" the magnetic field. AC current is applied which creates a wave for the fin to ride. LIM powered coasters use multiple sets of magnets on both sides to power their rides. An Example of a LIM Powered ride is Flight of Fear at Paramount's Kings Island and Kings Dominion.
LSMs: Linear Synchronous Motors use more basic magnet theory, attraction and repulsion. Strong, permanent, rare-earth (come out of the ground magnetized) magnets are attached to the train. On the track are electro-magnets. As the train approaches one of the track-magnets, the track-magnet is set to attract the magnets on the train, pulling the train forward. After the train-magnets pass over the track-magnet, the track-magnet is set to repel the train magnet, which pushes the train along. The are multiple sets of electro-magnets on the track and must be fired in sequence in order to propel the train to top speed. An example of a LSM powered ride is Superman: The Escape at Six Flags Magic Mountain.
Obscure physics knowledge.
http://www.thrillnetwork.com/ for the actual explanations, so that I can lazily not have to write it all out.