Fortunately - or not, depending on who you talk to - particle beam weapons are still in the early stages of development. Nikola Tesla published the first technical description of a charged particle beam in 1937 in an attempt to provide a superweapon that would put an end to all war, but he was never able to act on his plans. The main impetus for its development in recent years was the "Star Wars" defense program of the 1980s, which has since passed away, so the future of the device is somewhat in question. That's not to say that particle beams aren't in use - particle accelerators and prototype magnetic confinement fusion reactors use them all the time. There just aren't many military applications for them at the moment (outside of science fiction and computer games, that is.) Still, a certain amount of research into weaponized particle beams has been conducted, and may continue for some time.
Particle beam weapons work by accelerating a stream of atoms or subatomic particles to near-relativistic velocities and projecting them in (surprise) a beam. Both electrons and protons can be used to form this beam, and would be the choice for a weapon to be used within an atmosphere. Hydrogen atoms are the preferred choice for an extra-atmospheric weapon - they have a neutral charge, and thus the beam wouldn't be deflected by the earth's magnetic field, or scattered by the mutual repulsion exhibited by charged particles. Any type of particle beam would transfer a large amount of energy from the beam to any object struck by it, resulting in damage from the swift temperature increase and possibly an explosion. Think of the effects of a lightning bolt - which is essentially a charged particle beam - and you'll get some idea of how destructive such a weapon could be.
There are two main components to a particle beam weapon: a power supply, and an accelerator. A power supply is simply that - a mechanism that supplies the power needed by the accelerator to produce and accelerate the particle stream. The problem is that accelerators require a great deal of power - millions or even billions of watts - in a very short time. Thus, a very powerful generator is needed, along with a capacitor bank or other power-storing device to accumulate power from the generator and dispatch it to the accelerator in the short, powerful bursts necessary to create destructive pulses. The technology to create such a power source already exists; the problem, of course, is making it small and light enough to be portable - a particle beam is a direct-fire weapon that must have a line of sight to its target, so it would be of limited use if it were tied to one physical location.
The accelerator presents an entirely different set of challenges. A working accelerator requires three components: a particle source, and injector, and the accelerator hardware itself. All current accelerators work the same way - the particles from the injector pass through a series of devices, each of which applies an impulse to the beam until the stream has reached a sufficiently high energy to be useful. These accelerator modules are currently limited in the amount of power they can add to the beam - applying too great a force in one step tends to distort it, which is clearly undesirable for a device that relies so heavily on precise targeting. Thus, the accelerator path of high-power research colliders are very long; to produce a 1 GeV beam, you'd need a linear accelerator no less than 100m long, and probably more. This is clearly far too large for a weapon system, and no existing technology is capable of reducing the requirement.
The size issues of the weapon itself are not the only difficulties with the particle beam weapon concept - there is also the challenge of accurately targeting a very thin beam at long ranges and the propagation issues of a charged-particle beam in the atmosphere (lightning bolts are none too precise.) The US Air Force's evaluation is that a space-borne neutral particle beam weapon is not feasible by 2025, and intra-atmospheric weapons are not even under consideration. Thus, it is likely that this weapon will remain fiction for the foreseeable future. It's pretty cool fiction, though!
I used information from PBS' "Tesla: Life and Legacy," the USAF's "Space Operations" report for the Air Force 2025 project, and Richard Roberd's excellent "Introducing the Particle-Beam weapon" in the preparation of this writeup.