Terrain
Terrain in Civilization IV works largely the same as in Civ3: every tile has a base (immutable) terrain type (Grassland, Plains, Desert, Tundra, Hills, Mountains, Ice, Coast, Ocean) and on top of that Jungle or Forest may exist. Rivers flow between tiles (and not through them, as in Civ2, SMAC and the CtP series) and give any bonuses they may give to all tiles that border on it (but they do not give movement bonuses as in aforementioned games). Attacking across a river gives the defender a bonus. Civilization IV also features single-tile fresh water lakes [see image]. Proximity to fresh water (lake or river) gives cities a Health bonus. Certain terrain types give a bonus to defense or offense (Forest, Jungle, Hills) though not as many as in Civ3 and as in all Civ-like games the speed of Unit movement is affected by terrain as well. Spread throughout the map are 31 different types of Resources (Civilization IV: Warlords will add a few more), from Wheat to Iron and from Incense to Whales -- these have a major impact on gameplay [see below]. How these Resources are distributed over the map has been improved compared to Civ3, it's more balanced this time.
As in Civ3, players can build Workers to enhance the terrain around their cities [see image]: build Roads and Railroads to connect cities in a trade network and to speed up movement, build Farms to increase Food production, Mines to increase Hammer production. But besides these four traditional ones, Civilization IV supports many new types of Improvements, such as Windmills, Lumber Mills and Workshops. New in Civilization IV is also that certain Advances and Civics increase the output of Improvements. Every Resource also has a specific Improvement that is required to be built before Cities can take advantage of them: a Winery for Wine, Fishing Boats for Fish, Whaling Boats for Whales, a Quarry for Stone, etc.
The overall size of the entire map has been somewhat reduced compared to Civ3, this is the cost of switching to 3D. To compensate for the loss of epic map sizes, epic game speed has been introduced instead [see Game Speed].
Resources
In Civilization IV all Resources that can be found of the map have two uses. The first is the same one as in previous Civ games: they give a bonus to resource collection. Where a ordinary Plains tile gives 1 Food and 1 Hammer, a Plains tile with a Stone Resource and a Quarry (which is required to access it) gives 1 Food, 4 Hammers and 1 Gold. The other use is that if a City has a Resource in it's Resource box, it gets a certain bonus.
As in Civ3, in order for a Resource to be in the Resource box, it needs to be connected to it by the Trade Network of its owner. A Trade Network consists of all tiles that are connected with each other by Road, as well as all Cities that are located along a river or coast line. A Resource tile has to be improved with both its matching type of Improvement plus a Road in order to be part of a Trade Network. Resources can be exchanged between players, as long as the Trade Networks of both players connect with each other. Such deals can be arranged through diplomacy [see Diplomacy]. A player only needs one instance of a Resource for full benefit (assuming the Resource and all cities are part of a single Trade Network) but if a Resource is traded away to another player then the owner doesn't get the benefits him/herself anymore (nor can it be traded again to other players). If however (s)he has more than one instance of the Resource, then (s)he can trade each instance away to another player, and as long as (s)he keeps at least one instance for him/herself, (s)he gets the benefit him/herself as well.
As mentioned, every resource in the game gives a bonus. There are three types of bonuses available, or rather 3 types of Resources: Strategic Resources, Luxury Resources and Food Resources. For every type of Luxury Resource in a City's Resource box it gets a Happiness bonus and for every Food Resource it gets a Health bonus. Strategic Resources allow the production of certain items or make certain items cheaper (though the system is less restrictive than in Civ3). For instance Marble makes Wonders cheaper and Iron and Copper are required for the production of certain types of units.
Go to the Civilopedia for a detailed breakdown of all Terrain, Improvements and Resources.