Have fun! Yellowstone is pretty cool. If you write the park office they will send you information on wilderness camping, access to trails, how to deal with the wolves, water, bear and moose. You might know a lot already, but they have some good ideas anyway.
Almost regardless of where you go in August, in addition to the possibility of snow, you might run into people. You don't have to be high in altitude to get snow, and you don't have be low in elevation to have people tromp by your camp. Which is worse depends on your preferences. Just thought I'd mention both because I've had campers mention how they were surprised, and disappointed, by both.
Everyone I know calls it "Jellystone," too, from time to time. And we ask people questions about their trip adding the nicknames "Yogi" of "Boo Boo." If they're uptight sorts we call them "Mr. Ranger."