Madison Campground is one of the most popular campground in Yellowstone National Park, and for good reason. It is the closest campground to the geyser basin areas, which are themselves very popular, and it can accommodate most camping vehicles. It also is available for reservations, which should be made well in advance. In addition to all that, it is a very nice campground, with plenty of wildlife and great fishing in the Madison River nearby.
Madison is set up on eight loops, two of which are only for tent campers. The quality of the campsite depends a lot on where it is in the loops. The sites at the outside end of the loops tend to be larger and more private, but some of them are close enough to the road to get some noise. The sites along the main access road (#188 - #199) can be very nice, with great views and no one behind. Each loop has full bathrooms and drinking water but no showers.
While reservations tend to be hard to come by on short notice, there are almost always cancellations, and some sites come available every day, so check with the office when you arrive.