There are a few different types of laser cutters, but they all essentially use the same process of using a laser to cut material. The laser originates from a laser resonator, which sends out a beam of intense light through reflects through a system of mirrors to the cutting head. Within the cutting head, the laser is focused through a lens and narrowed down to an extremely thin, concentrated beam. This beam is projected down at the material and can cut or raster the raw stock, which I'll cover in more detail later. The cutting head is usually mounted on what is called an XY gantry, which is a mechanical system driven usually by belt or chain that allows for the precise movement of cutting head within a given rectangular area, which is the size of the work bed. The gantry allows the laser head to move back and forth and forward and back over the work piece so that it can make precise cuts anywhere on the bed. In order for the laser to actually cut, the focal point of the lens, where the laser would be at its finest, needs to be on the surface of the material it is cutting through. All laser cutters require a focusing procedure before making their cuts to ensure that the laser cuts well.
The difference between different types of laser cutters comes from what types of lasers the machines have. The type of laser defines what type and thickness of material it can cut through, because different laser types have different power ranges. Usually, higher power lasers are used on the industrial scale to cut out large sections of sheet metal or plastics, while lower power lasers are used for a wide range of thinner, more potentially flammable materials like paper and card stock, wood, and some plastics.