The river is a large, flowing body of water located within the forest biome. As of the new update, it replaces the lake as the only natural water source inside the forest. It can be anywhere in the forest depending on which server you play on, and stretches from the top of the forest to the bottom.
The river also has a current, which pushes the player downwards. However, if you just continue moving left or right, you will not be effected by it. Also, if you put a bridge, you will also not be affected by the current.
At the lake that represents the end of the river, the current stops.
While under the low quality setting, you cannot see the current. However, not only with the high quality setting you can see it, but also lilypads and presumably some rocks.