OK, so people say the 5D3 has a 1 stop improvement in RAW over the Mark II. However, what exactly is a stop.
A 'stop' is a doubling or halving of light throughput. The ISO number is simply a standardized value for film/sensor light sensitivity (which is called the film 'speed', which by the way has nothing to do with processing or focus speed). Doubling this ISO value means doubling the light sensitivity (and thereby effectively the light throughput). A 1600 ISO sensor value will be twice as light-sensitive as an 800 ISO value, and an 800 ISO sensor value is twice as sensitive as ISO 400, which in turn is twice as sensitive as ISO 200. That means that ISO 1600 is three stops more sensitive than ISO 200. 200 -> 400 (1 stop) -> 800 (2 stops) -> 1600 (3 stops). Three stops difference means it's 2*2*2 = 8x as light-sensitive. With the same aperture and exposure time, pictures can be taken with 8x less light at a three stop higher ISO value.
The same 'stops' are used for the light throughput of the aperture. An aperture value of f/2 lets in twice as much light as an aperture value of f/2.8. Another halving of light throughput in aperture is f/4, then f/5.6. That means that an f/2 aperture is three stops more light-sensitive (three stops "faster") than f/5.6. This again means that it is 8x as light-sensitive, which means that it can take photos at similar shutter speeds and ISO values with 8x less light.
The catch for higher ISO values is the introduction of "noise" (errors in the sensor's light recording, resulting in coloured or brighter pixels than they would be with a 'clean' exposure). Therefore an ISO value may not mean the same as usable ISO. For example, the 12800 ISO extension of the 60D crop camera is VERY noisy. It will not produce usable images for big or even medium-sized prints. At the same time, 12800 ISO with the 5D mk III will be very usable, with relatively low amounts of noise. The same ISO value here only means that the film is equally light-sensitive, meaning that exposure will be the same for the two cameras when at the same ISO, aperture and in the same setting. The amount of noise introduced is vastly greater for one than for the other though, so there's a big difference between the usability of these two cameras' equal ISO values.
In this case, a one stop of ISO improvement means effective (usable) ISO in RAW files. That means that the 5D mk III will do ISO 25600 with comparable success (noise amount) as one stop lower ISO (ISO 12800, half as light-sensitive) would do on the 5D mk II.