A pipe is just an object with a big hole into which to stuff pipe tobacco and a smaller hole from which to draw out the smoke from the bottom of the big hole. Pipes have been made of any material that can withstand the combustion temperature of tobacco: glass, ceramic, clay, mined mineral (like Meerschaum), the stone of various sorts (e.g. soapstone), many varieties of wood, some plastics (Dr. Grabow “Brylon” pipe bowls are made with nylon that has been blended with briar dust.), and of course, corncobs. Wood pipes, made from nearly any non-toxic hardwood, are available with and without the tree bark still on. The stems can be made of acrylic, hard rubber, bamboo, or any other suitable material that can provide an enclosed, stiff tube.
Most pipes found in a tobacconist will be made of briar root. This is the extremely hard root of a Mediterranean heather shrub.

Bowl and shank are ashwood.
Stem is bamboo, with a plastic tubing bit.
The matching tamper is also from ashwood.
Ash is unique in that the pith of the branches, both large and small, is soft and spongy. This can be cleanly removed by passing a hot wire or rod through it. This also means that the bottom of the bowl should be lined with a spackle of 50% plaster of Paris: 50% fine sand, to seal and fire-proof the pith there.

Hand bored from a home-grown cob.
The stem is a bamboo garden stake (undyed).
The soft bit is a section of Tygon plastic tubing.