Fog can affect units' reacting and detecting range when the enemy approaches. Fog also makes it more difficult for archers to aim and less likely to hit the enemies.

Snow and rain will reduce units' visibility, and affects archer's shooting accuracy as well.
Rivers will slow down units' movement when crossing, depending on the depth of the water. Soldiers move faster in the shallow areas then the deep areas. Forests can provide a massive tactical advantage. Units will move slower through forests, cavalry will have a more difficult time charging, and spearman formations are more vulnerable to being routed. Also, forests can be set on fire, allowing you to immolate your enemies.
Night battles will increase the chance to trick enemy generals and sow confusion among their ranks. Hills provide tactical advantages for archers and cavalry. Archers can shoot further when on higher ground, and cavalry can build up thier momentum faster when charging down from hills. Conversely, units will be slowed down and lose impact while charging uphill.