Leica R lenses are 35mm manual focus lenses that fit on Leica R SLR bodies. Most the lenses can be used on any Leica reflex camera body in a non-metered mode (they physically fit onto the body). There are different designations that denote the lenses cams, or cam-coupling, which is how meter-coupling is carried out. So in order for the lens aperture information to make it to the camera for proper metering, you must choose the appropriate lens and camera combination.

A '1-cam' or 'single cam' is for original Leicaflex cameras. The cam looks like a sloped and curved chrome bar located in between the lens mount and the rear element of glass. It is not recommended to put these lenses on R8 or R9 cameras because it can damage the cameras ROM contacts.
2 cam
A '2-cam' or 'twin cam' is for the SL and SL2 models and original Leicaflex. The 2 cams on these lenses also look like sloped and curved chrome bars in between the lens mount and the rear element. These lenses are also not recommended to be put on the R8 or R9 cameras. (The 1 and 2 cam lenses can be used on other bodies in stop-down metering mode only.)
3 cam
A '3-cam' or 'triple cam' will work on all Leica SLR models. The 3 cams on these lenses include the 2 cams previously mentioned (the 2 curves chrome bars), plus a 3rdcam that is black, has a 3-tiered step shape, and is located inside the 2nd cam.
3rd cam
close up of the 3rd cam

A '3rd-cam' or 'R-only cam' is for R3 and later models. This lens does not have the 1st or 2ndcam (the chrome bars) at all. It only has the 3rd cam (the black tiered cam) located in between the lens mount and the rear element. These lenses also have a slightly different mount shape that prevents it from fitting on earlier camera models (Leicaflex).

ROM (with 3rd cam)
ROM means that the lens is supplied with electrical contacts. These lenses are for use with the R8 and R9, although they are still compatible with earlier models but without the extra benefits from the contacts. This lens is also a 3rdcam lens, but has a strip of electrical contacts in between the mount and rear element. The contacts allow additional information to be communicated between the lens and the camera, such as the focal length of the lens for flash metering. 
* In the KEH descriptions, we use the terms 2 cam, 3 cam, 3rd cam, and ROM. You can shop the Leica R category here.