Game Info
Some buildings are only available when the city is built next to a river (or a lake). They are:
Strategy
Rivers are sources of fresh water, allowing the construction of Farms on adjacent hills and tundra. Farms next to fresh water receive an additional +1 
Food Food when the Civil Service technology is researched. Units expend all of their movement points when crossing rivers. This is true even if the tiles are connected by roads, until the civilization learns the Engineering technology. Finally, units receive a -20% penalty when attacking over a river.
Brave New World
As of Brave New World, rivers no longer provide an initial 
Gold Gold bonus.