The person who made parchment was a parchmenter. He would first have to acquire the raw material, which probably meant a trip to an abattoir to choose the best skins for use. Assuming they were completely unprepared, he had to gauge how the color of the fur indicated the final color of the parchment. White fur meant you would have lighter pages when all is said and done, but darker colors were probably more common. The skins might have holes in them from damage the animal did to itself or from tick bites.
The first step in preparation was to wash the skins in cold water. Skins were then commonly soaked in vats of water with lime for three to ten days, stirring regularly, after which they were laid out so they could be scraped to get the fur off. They were soaked in water for a couple more days after the scraping. The now waterlogged skin was stretched on a frame to dry flat. Strings attached the skin to pegs on a wooden frame; the pegs would be tightened to stretch the skin. You would not want to put holes in the skin for the strings, however, so the edge of the skin was rolled around pebbles and the end of the string tied around them.
Once dried, the skin was now scraped thin and smooth with a lunellum, a blade shaped like the crescent-moon so there was no sharp point that would accidentally pierce the skin. Rubbing with chalk would help to further smooth the result. Pre-13th century examples of parchment were thicker than later sheets, as the process was used to make thinner and thinner pages. These pages could then be written on and cut to be bound.
The development of parchment allowed books to last in good condition far longer than paper. So what's the oldest (medieval) book in good condition, and why was it in the coffin of a 7th century saint? For that story, you'll have to stay tuned.