There's also the North Rim, which is now contracted out to Forever Resorts (used to be Xanterra):
http://www.grandcanyonforever.com
Of course the South Rim is the more popular spot. The main concessionaire is Xanterra. They operate Maswik Lodge, Thunderbird Lodge, Kachina Lodge, Bright Angel Lodge (and cabins), and the El Tovar Hotel.
http://www.grandcanyonlodges.com/lodging/
The El Tovar is the nicest accommodations and dates back to when it was under US Forest Service jurisdiction. The exterior was used in National Lampoon's Vacation. It's a prime example of the "national park rustic" style.
Maswik is more basic accommodations with a lodge building and separate buildings for the rooms. When I stayed there they still had budget quad cabins - but since removed. I paid $80 a night for three people about a decade ago. They weren't fancy by any means, but each room had a fan, a TV, and a phone.
Thunderbird and Kachina are kind of odd. They're designed them in a modern 60s style that looks kind of out place in a national park. They were part of the Mission 66 plan to build new NPS facilities in a more modern style rather than the "national park rustic" style that is favored today. Nothing wrong with staying there, but they look kind of odd in a national park. But if "chain hotel" is fine, they frankly feel like that - kind of like Las Vegas or Palm Springs looked like in the 50s/60s.
Bright Angel Lodge has these cute little cabins. Many of them are within maybe 50 feet from the rim.
Delaware North now operates Yavapai Lodge and the Trailer Village. I believe that includes the market. Yavapai Lodge is more basic motel style accommodations.
https://www.visitgrandcanyon.com