A vinyl cutter is a computer peripheral used for the creation of vinyl signs, decals, banners, vehicle decorations, numbers on the shirts of ball players, etc.
The vinyl cutter works just like a plotter. Indeed, it is a plotter with some added functionality.
Instead of a plotter pen, a vinyl cutter uses a cutting blade. The vinyl cutter contains controls which let you choose how deep into the material the blade should cut, and how fast/slow it should proceed.
Since the vinyl cutter is a plotter, you can use plotter pens to plot the result on paper and examine it before committing the design to vinyl (which is more expensive than paper).
Once you have cut the vinyl, you use a weeding tool to weed out the background (i.e., the parts of the vinyl that do not belong to the design).
Finally, you transfer the vinyl to wherever you want it: The back of a T-shirt, the bumper of a car, and such.
The original design starts as a vector graphic created either by drawing software (such as CorelDRAW!), or CAD. Bitmap graphics are not suitable for this purpose unless they are first converted to vector drawings.
The latest models of vinyl cutters appear to the computer as just another laser printer (with an appropriate driver), so any drawing software can send its output to them. Or, for that matter, any software. Sign makers often use special purpose software which is very expensive and copy protected. Typically, the only thing it can do that regular off-the-shelf vector graphics software does not is draw a box around the design. This makes weeding much easier.
When I worked for a sign business, I did all my designs in CorelDRAW!, and just drew the box myself. Duh!