Rivers do not run in squares: instead they run along the edges of squares.
Any square with a river along its edge produces one extra Commerce if worked by citizen laborers within a city radius. Also, rivers are sources of fresh water, allowing irrigation of adjacent squares without Electricity.
A city that is built adjacent to a river (or another source of fresh water) does not need an Aqueduct to support a population above 7 (but still has a maximum cap of 12).
The value of a clean source of fresh water to a population cannot be underestimated, and river banks have always been prized sites for settlements. In addition to fresh water for drinking, irrigation, and waste removal, rivers are valuable trade routes and have been used to aid the defense of the settlements on their banks.