Vellum is a translucent animal hide product, made from goat or sheep, usually. It is thin - about as thick as paper, and is used a writing surface, and for reinforcement, support, and cover in bookbinding. As it is an animal hide product, it is expensive. Unlike leather, vellum's unique properties make it relatively easy to reuse.
Vellum is strong, resists abrasion, and is extremely flexible, even after a long time. It's biggest flaw is that it tears relatively easily. The majority of books and documents in Europe through 1450 were written on vellum.(1) As documents and books were destroyed, or large portions of them lost, the remaining fragments were used to reinforce and make new books. From time to time, important early manuscript fragments are found when very old books are torn down to be rebound. For instance, I can see through the endpapers on my copy of P. Vergili Maronis Codex Antiquissimus that it is sewn on vellum tapes cut from a early to mid 15th century manuscript, with rubricated initials in red and blue.
Someone now might want to reuse antique vellum for the aesthetic properties of it - the script used in many 19th century legal documents looks interesting. Or one might reuse antique vellum due to the cost - it is much much much cheaper than buying new vellum. The cheapest new vellum costs about $10 a square foot, and can be as much as $25 a square foot. Nineteenth century legal documents on vellum can be readily found on eBay for $0.50 - $1 a square foot.
Many librarians and historians argue that it is irresponsible to destroy 150 year old documents just to save money. I have a hard time disagreeing with that statement, though I countinue to cut these legal documents up. If I didn't, they'd just end up framed on someone's wall or lost, or so I claim. Your call.
Reused vellum can either be used as a structural element of a book or as a writing surface. (2)
Vellum can be used for bookbinding for tapes, reinforcing the spine, or for lining the edges of the boards, to prevent abrasion, without any modification. If it is to be used for the binding of a book, it can either be used with the previously existing text visible, or with it hidden, or with it completely removed. If the boards that are used for binding are not white, a thin layer of paper should be glued on them first, as the brown of the board will show through the vellum. To remove the text from a piece of vellum before using it for bookbinding, see below.
Vellum is a wonderful surface to write on - it is so smooth and durable. The easiest way to reuse it is to find a document that is thick and only has text on one side, and just reuse the other side. The other way is to scrub the text off.
To scrub the text off, soak the vellum thoroughly in water. Remove, and use a Scotchbrite™ or similar green plastic scouring device to gently scrub the text off. If necessary, apply additional force. If this does not work, a razor may be used to scrape off the remaining text, though the resulting texture is usuallly not desirable. Keep the vellum wet, as needed.
Once the text is removed to your satisfaction, blot the excess water off with blotter paper. Using a staple gun and staples long enough that they do not go all the way into a target piece of wood, staple the vellum to it, so that some air can get behind it. Staples should be along the edge, puncturing as little of the usable vellum as possible. The vellum should be straightened, and not too taut, as it can tear out of the staples while drying. Once the vellum is dry, it will be a nice surface for writing on or bookbinding.
1. See legbagede's wonderful Aristotle's Lost Library, Medieval Andalusia & Chinese Paper, or How Europe Learned to Learn Again and Why the Renaissance Happened When & Where It Did and Historical Evidence Regarding the Libraries of Muslim Spain for a more full explanation of this.
2. Or for any number of random things, of course - I used little scraps left from bookbinding in collages.