First thing I want to address is size. I don't want anyone having any misconceptions about snapping turtles, they get to be around two feet from head to tail when fully grown. Alligator snapping turtles can get to be around three feet in length.
The other answer that states they only grow to eight inches was based on the article that was talking about their sexual maturity. Snapping turtles will start mating once they reach that length, it doesn't mean they will stop growing.
Snapping turtles are not pets for people with limited space.
While it is only a couple inches in length, you can get by with housing it in a 10-20 gallon aquarium, but once it surpasses eight inches, you should be keeping it in a 55 gallon. For an adult snapping turtle, the minimum size of an enclosure should be about 2 feet deep and 4 feet long.
The depth of the water should be the same as the turtle is long, so the height of the tank should reflect that. Also note, since you will probably need to build the enclosure yourself, you will want to account for water pressure being applied to the tank as you build it.
Snapping turtles don't normally rest above the water as other turtles do, but it's still a good idea for you to place a rock or something large on one part of the tank, that way the turtle can rest with it's head above the water if it's feeling ill.
Important note: Snapping turtles are great climbers, and have been known to easily scale fences that people foolishly place in their way. Be prepared to find the turtle wherever it wants to be in your house.
Lighting is the same as other turtles. You should provide some UV lighting and a heat lamp to one side of the tank for basking in.
You can feed them commercial pellets, but I would suggest also offering them a variety of other things, such as live shrimp and fish (such as goldfish or guppies), cooked chicken or turkey, and nightcrawlers. Also make sure to feed them some vegetation like mustard greens or water hyacinths.
Finally, be prepared to never touch the turtle. Snapping turtles have the one of the strongest bites in the entire animal kingdom, do not get any body part near their mouths. Snapping turtles are well known for taking fingers and toes of people who fish for them. Or of people who just happen to have been near them.
Snapping turtles are not pets for people who want to touch their pets.
If you need to move the turtle for any reason, you should pick it up by the tail, and provide some support with you other hand at the bottom of the shell. Be especially careful, as alligator snapping turtles might have some difficulty turning their heads around to bite you, but common snapping turtles have no problems with reaching nearly all the way back to their tail.