The Chronicle has a bunch of articles on the annual US News ranking of US universities. I don't know how long they will be available before going behind a paywall, so get them fast.
The US News happens to be the biggest such ranking exercise, even though it has come in for some serious criticism (and ideas for improving this exercise are available). If you look at the criteria used by the magazine, you realize that they are heavily skewed against the public universities. Result? The top public university (UC-Berkeley) in their list is at Rank 21!
We asked ourselves: What are reasonable indicators of how much a school is benefiting the country? We came up with three: how well it performs as an engine of social mobility (ideally helping the poor to get rich rather than the very rich to get very, very rich), how well it does in fostering scientific and humanistic research, and how well it promotes an ethic of service to country. We then devised a way to measure and quantify these criteria (See "A Note on Methodology"). Finally, we placed the schools into rankings. Rankings, we admit, are never perfect, but they're also indispensable.
The other one is in this very interesting site which allows you to get your own personalized ranking of graduate schools in your field based on criteria you choose. We looked at this site a while ago (here).
Evidently, these other methods have not entered the public consciousness in the US the same way the US News's has.
Before I finish, I just want to point to Brayden King's discussion of how universities (organizations, in general) deal with rankings in which they don't fare well.