The ram accelerator is a method of launching a projectile from a fixed mechanism. That's a lot of words for 'it's a gun.' What makes this gun special is how it works. In a 'regular' gun (and we're talking idealized 'gun' here) a propellant is ignited behind a projectile. The propellant expands rapidly, creating high pressure behind the projectile which is forced along the barrel by the pressure differential. The problem is that this isn't a very efficient way of doing things, in terms of the energy extractable from the propellant. As the projectile moves, the volume behind it increases; this means that as the projectile accelerates, the amount of energy transferred from the propellant to the projectile begins to drop as the difference between its velocity and the ideal expansion velocity of the burning propellant decreases. In essence, the lowest pressure area in a standard gun is the spot right behind the projectile - which is where you want the highest pressure.
The ram accelerator is essentially a decoupled ramjet. In this model, the gun barrel (the accelerator) is filled with a mixture of gaseous propellant and then sealed. The projectile, which is a subcaliber one and hence will allow gas flow around its edges, is fired into the sealed barrel by an 'initiating' gun mechanism, at a precalculated speed. It is followed by a disc called an obturator intended to produce the initial shockwaves and ignition in the propellant. This disk swiftly falls behind the projectile due to its larger cross-section. The projectile creates shock waves around its perimeter as it flies, and it plus the surrounding barrel act like a ramjet - the compression of the propellant gases causes them to ignite, creating a high pressure area immediately behind the projectile. As the projectile travels down the barrel, it 'carries' its high-pressure area with it. It is notable that the projectile can (and has been observed to) accelerate to velocities greater than the detonation speed of the propellant gas. This is because, as in a ramjet, the flow past the projectile is 'decelerated' to subsonic rates by standing shock waves, so that the actual combustion is occuring in a subsonic regime and transferring energy as it goes - in other words, the projectile isn't just being pushed by the standing propellant burn, it's using the surrounding propellant as an engine would. There appears to be indications that a scramjet-like regime can be achieved as well, pushing theoretical projectile speeds even higher. This is one reason for having 'staged' propellant sections, though; by having the propellant enter a regime with a higher standing pressure, the Mach number for that regime will increase in relation to the 'outside world' and the projectile can continue to operate in a subsonic combustion mode relative to its environment (the speed of sound increases as the pressure increases).
The ram accelerator was conceived and initially built at the University of Washington beginning in 1983; using a 16-meter accelerator, projectiles were accelerated up to 1600 m/sec. over the length of the accelerator. In other units, they did the equivalent of zero to 3,579 mph over that same distance. Or for another comparison, the U.S. Navy's Mk. 45 5"/54 caliber naval gun mount, as carried on the DDG-51 Arleigh Burke class guided missile destroyer among others, accelerates a shell from rest to 800 meters per second over a 9-meter run to give the 70-lb (32 kg) shell a range of 13 nautical miles. Although that's on a close order as the ram accelerator's performance, the difference is that the ram accelerator could theoretically be extended in length, and that additional length would all be productive in increasing the projectile's velocity - unlike the cannon, which has a maximum length not much longer than the optimum size where any further extension actually slows the shell down.
For more information on the UW ram accelerator, see: http://www.aa.washington.edu/AERP/ramac/index.html