Terrain is "the physical features of a tract of land" (m-w.com). In EU2, each province has one terrain type that is assumed to cover it uniformly. Terrain has a strong effect on the movement and combat of land units.
There are five types of terrain: desert, forest, mountain, plains, and swamp. In addition, rivers exist between some provinces. The terrain is the same in all scenarios, being defined in the file province.csv. You may find variations in some mods.
On the map, desert is yellow, with little "palm tree" icons. Desert requires 12.5 days of movement for cavalry to enter. It offers the chance to get the cavalry bonus in combat. In some cases, desert may be white in color, looking like plains, i.e. Egypt. To completely distinguish plains and desert requires observing the tooltip for attrition found by hovering the mouse over the skull icon in a unit's details window. The unit must be in the province. If maximum attrition shows +5% for desert, then it is a desert province.
On the map, forest is green with a yellowish cast, with little "tree" icons. Forest requires 12.5 days to enter. Forest reduces cavalry shock and firepower by 50%.
On the map, mountains are orange, with little "peak" icons. Mountains require 17.5 days of movement for cavalry to enter. Mountains are the best defensive terrain, offering the defender a +2 DRM. Mountains reduce cavalry shock and firepower by 50%.
On the map, plains are beige, with no icons or occasional "low hills" or "clump of trees" icons. Plains requires 10 days of movement for cavalry to enter. It offers the chance to get the cavalry bonus in combat.
On the map, swamp is green with a bluish cast, with little "low vegetation" icons. Swamp requires 15 days of movement for cavalry to enter. Swamps slow down combat, applying a -1 DRM to both the attacker and the defender. Swamp reduces all combat unit types' shock and firepower by 50%.
On the map, a river is shown by a blue line between two provinces. Rivers require an extra 5 days of movement for cavalry to cross. Attacking across a river is hazardous, giving a +2 DRM to the defender.
A strait is shown by a body of water with little "pebble" type icons between provinces. Veneto is connected to the rest of Europe by a number of straits although the graphics do not represent the straits. Straits require an extra 10 days of movement for cavalry to cross. Attacking across a strait is hazardous, giving a +2 DRM to the defender. Straits may be blocked by any number of non-retreating enemy warships in the sea zone which the strait crosses. (Transports cannot block straits.)