Terrain feature found everywhere across the world. Unlike other features, it doesn't occupy a separate terrain tile, but extends between several tiles. A river usually starts next to a mountain tile, and it always ends in a larger water tile, such as a lake or ocean. A river will also turn adjacent flat desert tiles into flood plains, adjacent snow tiles to tundra tiles, and adjacent tundra tiles to plains.
Traditionally, cities have been built along rivers, and for good reason. Rivers provide a steady source of fresh water for both drinking and irrigation, improving the farmland around the city - Farms can be built in hills and tundra adjacent to rivers, and receive an additional +1 Food early on (whereas all other Farms have to wait until the Industrial Era). A city located on the banks of a river can also construct a number of special buildings...such as the Hydro Plant, which adds +1 Production to all tiles next to the river!
Rivers also protect a city, as it is quite difficult to mount an assault against a city across a river. Units expend all of their movement points when crossing rivers - even if the tiles are connected by roads - until bridge-building techniques are discovered, and suffer a 20% penalty when attacking units on the other side of a river. If you control the banks of a river, you may gain a very important tactical advantage over an enemy.
As of Brave New World, rivers no longer provide additional Gold yield for neighboring tiles. However, they become incredibly important for trade, which is reflected by the additional Gold bonus when opening international trade routes to cities built on rivers.
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 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 they are attacking across a river.