Rivers are a kind of terrain. They offer no defensive cover, and are hard to cross. In the Advance Wars games, they can only be crossed by Mechs, Infantry and Air Units. In Famicom Wars and Game Boy Wars 1/2/Turbo, Main Battle Tanks and APCs can cross them, but require all of their 6 Movement Power. In Game Boy Wars 3, they could be traversed by cars (3 Movement Power for both types) and any vehicles with the River Treads Movement Type (4 Movement Power).
Although the Rivers themselves can't be crossed by most other Units, they may have a Bridge spanning them which allows the other units to pass through. In Game Boy Wars 3, the Constructor could build a Bridge on a River tile to handle this.