Brian Dorney at CERN:
Firstly, physicists rely on a principle many of us learn in our introductory physics courses, the Lorentz Force Law. This result, from classical electromagnetism, states that a charged particle in the presence of external electric and/or magnetic fields will experience a force. The direction and magnitude (how strong) of the force depends on the sign of the particle’s electric charge and its velocity (or direction its moving, and with what speed).
So how does this relate to accelerators? Accelerators use radio frequency cavities to accelerate particles. A cavity has several conductors that are hooked up to an alternating current source. Between conductors there is empty space, but this space is spanned by a uniform electric field. This field will accelerate a particle in a specific direction (again, depending on the sign of the particle’s electric charge). The trick is to flip this current source such that as a charged particle goes through a succession of cavities it continues to accelerate, rather than be slowed down at various points.
A cool Java Applet that will help you visualize this acceleration process via radio frequency cavities can be found here, courtesy of CERN.
Now that’s the electric field portion of the Lorentz Force Law, what about the magnetic? Well, magnetic fields are closed circular loops, as you get farther and farther away from their source the radii of these loops continually increases. Whereas electric fields are straight lines that extend out to infinity (and never intersect) in all directions from their source. This makes the physics of magnetic fields very different from that of electric fields. We can use magnetic fields to bend the track (or path) of charged particles. A nice demonstration of this can be found here (or any of the other thousands of hits I got for Googling “Cathode Ray Tube + YouTube”).