For convenience, I've divided the rappelling (a.k.a. abseiling) devices in my collection into a number of categories. Although there are more rigorous methods to classify rappelling (abseiling) devices, I've chosen an informal approach that I think will be easier for most people to follow than some purely academic method. The categories are loosely defined as follows:

The rope follows an "S" path around two bollards. They come in four types:
Carabiner Methods
Rappels using carabiners only or carabiners and brake bars only.
Figure Eights
Devices that look like an "8."
Fixed Multi-bar Devices
The rope snakes around at least three bollards, all of which are fixed.
An ill-defined catch-all category, but generally the rope wraps around a rod.
Another ill-defined "catch-all" category for devices with horns or prongs.
Lever Boxes
Devices with an enclosed rope channel and a control lever.
Flat or slightly bent plates with holes and/or horns, not classified as Figure Eights, Hooks, or Horns.
Poly-bollards (the best name I could think of)
The rope snakes around at least three bollards, at least one of which moves to provide a stop function (otherwise, I'd call it a Fixed Multi-bar Device).
Single Brake Bar
Descenders have one and only one brake bar that swings open for rigging.
Devices with frames that accept a number of brake bars, at least some of which can move on the frame.
Devices where the rope wraps around a drum. The drum axis can be horizontal or vertical.
Squeeze Brakes
Descenders thatcreate friction by squeezing the rope between plates rather than bending the rope.
All the rest.

A word on the tables…

The tables include some numerical data:

 ID  This is just my catalog number so that I can keep these straight.
 Weight  Weights are in grams. Webbing, slings, etc. are not included.
 Height, Width, Thickness  I've given the dimensions in millimeters. The measurements are in perpendicular directions. I've chosen to measure the maximum dimensions instead of the most obvious dimensions. Sometimes this leads to numbers that are more than what you would expect - for example, the thickness of a bent plate would be more than the thickness of the unbent plate.
Standard Volume   The standard volume is just the product of the height, width, and thickness divided by 1000. This gives a volume in milliliters of a box that the device will fit into. Odd shaped devices are penalized by this formula, but since they are generally harder to pack, this number might be useful.