Types of Stones

The stones are usually made of the following materials, but can be composed of other things as well:
The earliest markers for graves were natural fieldstone, some unmarked and others decorated or incised using a metal awl. 
Granite is a hard stone and requires skill to carve by hand. Modern methods of carving include using computer-controlled rotary bits and sandblasting over a rubber stencil.
Marble and limestone:
Both limestone and marble take to carving well. Marble is a recrystallized form of limestone. Marble became a popular material from the early 1800s although its extra cost limited its appeal. 
Sandstone is durable yet soft enough to carve easily. Some sandstone markers are so well preserved that individual chisel marks can be discerned in the carving, while others have delaminated and crumbled into dust. 
Slate can have a pleasing texture but is slightly porous and prone to delamination. It takes lettering well, often highlighted with white paint or gilding.