Real parchment (i.e., "parchment", rather than "parchment paper") is made from animal skins. It is untanned, so is rawhide rather than leather. What distinguishes parchment from other forms of rawhide are that it is dehaired with a lime solution, and dried while stretched. It is scraped and scrubbed, to give a suitable surface for writing on.
The definitions of such animal hide writing surfaces have changed over time. At times, "parchment" has referred to such material made from sheep skins, and goat skins (goats often being considered honorary sheep); a competing and often finer product was vellum, made from calf skins. However, we see in Shakespeare's Hamlet Horatio replying to Hamlet's question "Is not parchment made of sheepskins?" that calf skin is permissible: "Ay, my lord, and of calves' skins too." Modern usage tends to be to classify the thicker coarser ones as parchment and the finer ones as vellum. Partly, this is a convenience because it can be difficult to identify the animal used. From the 12th century De diversis artibus, we have a detailed recipe for goat skin parchment:
Take goatskins and stand them in water for a day and a night. Take them and wash them until the water runs clear. Take an entirely new bath and place therein old lime and water, mixing well together to form a thick cloudy liquor. Place the skins in this, folding them on the flesh side. Move them with a pole two or three times each day, leaving them for eight days (and twice as long in winter). Next you must withdraw the skins and unhair them. Pour off the contents of the bath and repeat the process using the same quantities, placing the skins in the lime liquor and moving them once each day over eight days as before. Then take them out and wash them well until the water runs quite clean. Place them in another bath with clean water and leave them for two days. Then take them out, attach the cords and tie them to the circular frame. Dry, then shave them with a sharp knife after which leave for two days out of the sun. Moisten with water and rub the flesh side with powdered pumice. After two days wet it again by sprinkling with a little water and fully clean the flesh side with pumice so as to make it quite wet again. Then tighten up the cords, equalise the tension so that the sheet will become permanent. Once the sheets are dry, nothing further remains to be done.
There are similar products that can be tanned. Klaf/qelaf as used for some Jewish religious/ritual writing was originally tanned (the oak galls used contain tannin, from with our term "tanned" comes from), but the same name is often used for an untanned product (i.e., a form of parchment).
Parchment being a type of rawhide results in some important differences from paper. It is typically more rigid, less porous, and more expensive. The expense accounts for the common practice of recycling parchment by scraping off the original writing; the first two points made this practice much easier than with paper (ever tried erasing ink from porous paper?).
For more on parchment, see
Clarkson, C. (1992). "Rediscovering parchment: The nature of the beast". The Paper Conservator, 16(1), 5-26.
Ryder, M. L. (1964). "Parchment—its history, manufacture and composition". Journal of the Society of Archivists, 2(9), 391-399.
For photos of various types of parchment, see https://www.leather-dictionary.com/index.php/Parchment (which also shows some materials not usually described as parchment, such as rawhide lamellar armour).