There is some interesting historical information in Wikipedia in the article Red River Valley (song)

See also this "disambiguation page" at Wikipedia on the subject Red River.

As a Minnesotan, I am most familiar with our Red River, also known as the Red River of the North, which forms the boundary between Minnesota and North Dakota, and continues northward into Manitoba, Canada, finally emptying into Lake Winnipeg.

Notable cities along its route are

Wahpeton, North Dakota
Breckenridge, Minnesota
Fargo, North Dakota
Moorhead, Minnesota
Grand Forks, North Dakota
East Grand Forks, Minnesota
Winnipeg, Manitoba

That area suffered serious flooding in 1997.