Editor's Note: This is exactly what you think it is, all factual and detailed. If you have a weak stomach for such things, do not read any further. You have been warned.
This is perhaps the most important thing that I learned in high school (at least the most important thing I learned from classes). I learned how to do this while taking an advanced biology course, and the method I used was so successful that my teacher would bring me into his other classes so I could show them how to do it. To do this you will need:
After you've skinned the animal, you'll want to remove the muscles of the dorsal neck and expose the topmost cervical vertebrae (i.e. where the neck bones meets the skull in the back.) Now orient your shears parallel to the animal's shoulders, open the shears slightly and place the tips of the shears just on the sides of the top-most vertebrae; what you are going to do is sever the animal's spinal column by closing the shears -- but you must be careful! If the tips of the shears penetrate too far into the flesh of the animal, you will ruin the brain tissue located at the lower brain stem.
Now read carefully: carefully close the shears, cutting through the top cervical vertebra with a crunching sound. Don't remove the shears yet -- you must now separate the skull from the column completely, and you do this by twisting the shears on their long axis. This forces the skull and vertebral column apart, and prepares the animal for the next step.
The animal's head can now be pushed ventrally (to its chest), exposing the separated base of the skull. Slip the tip of your shears beneath the center of the base of the skull (this time your shears are parallel to the animal's spine) and make a small cut towards the front of the skull. Always use small cuts, as this reduces the likelihood of the shear tips penetrating the surface of the brain. Advance your shears carefully and repeat; after each small cut, twist the shears along their axis as before: this forces the two hemispheres of the skull apart. Continue along the center of the skull until you've reached the sinuses in the front of the animal's head (at this point, your twisting action should have made the opening in the skull quite large.)
Now you have made a large opening in the skull down the center of the cat's head. To finish the job, you will make similar openings along each side of the animal's head. You can start at either the front of the rear of the skull. I prefer the front, because it can be difficult to make the initial lateral cut from the back (without stabbing into the brain and ruining the tissue.)
So, with your shears again parallel to the animal's eye-line, make small steady cuts around the side of the head towards the back, again using a twisting motion to widen the opening in the skull. Once these cuts have been done on both sides, the hemispheres of the skull will be completely separated from the base of the skull and from each other. They will hold themselves fast to the brain, so I recommend removing them one at a time by using a firm but gentle pulling motion from one of the corners of the skull fragment. If you use enough force, both skull fragments should peel off with a loud snapping sound, and the brain will be visible.
To free the brain from the skull base, use your scalpel and probe to gently separate the brain from the inner skull surface and then lift the brain out with your fingers. If your initial cut was made in the right place, you should be able to see the complete cerebellum and the brain stem (as well as the complete top of the spinal chord if you're lucky), and you should be able to sever the brain from the remaining spinal chord with a simple cut of the scalpel.
And there you have it. If you perfused your animal with formalin you should be able to keep the brain in the open air for several days. Put it in a formalin-filled jar for long term storage.
Oh yes and the other two cats Mwuhahahaha: Repeat for two more animals -- (find some excuse, pretend you're helping a friend, whatever) -- and then when your instructor has left the room, go to the front of the class and begin to juggle the brains like a mad mad clown in a macabre cadaver circus. Do this for as long as possible. If people have to leave the room to vomit, don't worry about it, keep juggling. THIS is the real reason for learning to remove the brain; I guarantee that your classmates and friends will remember the time you juggled cat brains until the day they die.