Mesa Verde National Park is pretty remote, and there is only one campground option in the park. There are several places to camp outside the park, but it is highly recommended to stay at Morefield Campground if possible. The drive from Morefiled to the major sights in the park ranges from 45 minutes to more than an hour, and staying outside the park would add significantly to the drive times. That being said, Morefield is quite a nice campground in its own right and can make for a very pleasant stay.
The campground is spread out in four access roads (plus the group area) over a huge acreage. Unlike many national park campgrounds, this one takes advantage of the large amount of space available and has many huge sites that are well-spaced from any neighbors. The Pueblo Road accesses mostly tent or small trailer sites; the Ute Road accesses the full-hookup sites as well as some tent sites; the Hopi Road has a mix of large RV sites and tent sites with over 100 sites total; and the Apache Road also consists of large RV sites and tent sites.
The downside of this campground is that it is so large that the private concessionaires don't seem to even try to maintain it very well. The restrooms are old, and many of the sites are overgrown with weeds and need leveling and resurfacing badly. The Wi-Fi only works in sites that are within about 50 feet of a restroom, and cell reception in the valley is weak to none. On the plus side, besides its relatively convenient location, there is a nice store, free showers, a laundry, the cheapest gas in the entire area, and a cafe that serves an all-you-can-eat pancake breakfast every day. Despite the fact that it is set in a valley between mesas, the scenery is very beautiful, and the wildlife (mule deer and wild turkeys mostly) is plentiful. This campground, while a bit tired, is a wonderfully remote and relaxed place to camp.
Note that the few full-hookup sites are very popular, so it is best to make reservations early for these.