Strictly speaking, "brain tanning" isn't tanning. Hides and pelts can be preserved though by a variety of methods. I always preferred soap, simple homemade lye soap, or store bought soap,,, it's not so icky as brains. Pelts or hides can be simply soaked in water until the hair slips, but an addition of a small amount of lye or a large amount of hardwood ashes to the water will hurry things along. Of course, if you're leaving the hair or fur on don't soak the hide or pelt. What you're doing is pushing the natural grease out of the hide/pelt and replacing it with something else, but even when you've done that, the work is not finished. It would be hard to give a step by step in this format, but there are lots of lengthy treatise(s) written on the subject, and free on-line.
There are literally hundreds of recipes for preserving animal skins, hair on or off, but the thing you'll want to remember is that IT IS A LOT OF WORK,,,! I've read that the main ingredient is a liberal application of elbow grease. Anyone can do it, but it's work, frustration, failure, and more work.