There are many campgrounds. the link below lists them. Fishing Bridge is the only campground that offers full hookups. Grant and Canyon campgrounds offer showers, but no hook ups. Reservations are required. Check out the method of reservation at least 6 months in advance.
http://www.nps.gov/yell/planyourvisit/camping-in-yellowstone.htm
I stayed in Norris this past summer. It was first come first serve. We got there before 10:30 and were nearly too late. We did get a site, though. The following days we were always the first there

so paid one day at a time. There were no showers and the only electricity was an outlet in the bathrooms.
Norris is fairly centrally located and that is a benefit as the park is large. We did not realize the benefit of this until we started exploring the park. Waking to the smell of the Norris geysers was nice too.
Having been to Yellowstone before I would choose Tower Falls campground as it is in the mountains and feels alpine.
Norris was a very good second choice due to location and shade and a creek running nearby. Buffalo too.
I drove through Indian Springs but found it unappealing.
Mammoth Springs campground lacked shade.
All that said, I would have been happy to get any site in Yellowstone. The campgrounds are very quiet during the day because everyone is out playing. All we did was sleep in ours.
Good luck
Pic of our Norris Campground Site:
