Religion
Religion has been a factor in most Total War games so far. However, its importance has changed over time. Whereas it seriously affects the gameplay in some games of the series, it's a minor factor which barely matters in others.
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Medieval Total War
Rome: Total War
In Rome: Total War religion plays a rather minor role. The factions can build various temples or shrines to different gods. These temples grant bonuses which are in accordance with the role of the respective god. Whereas gods of war tend to give military based bonuses, other gods or goddesses can for example make a region's farms more effective or will improve the health and happiness of the local citizens.
Rome: Total War: Barbarian Invasion
In Rome: Total War, all factions all can build monuments to their gods which only affect the happiness of a settlement. However, in the Barbarian Invasion expansion religion has been reworked entirely.
Religion has become extremely important in people's lives. In the early days of the Empire, religions could quite easily co-exist peacefully but times have changed and religious unrest is now very common.
There are three religions; Christianity, Paganism and Zoroastrianism. Most characters and nearly all generals have a religious belief. Religious conversion rates are effected by characters and buildings in the given region, and by the official religion of neighboring regions.
Each settlement has an official religion, and each faction has a state religion. The state religion is determined by the faction leaders religion. The settlement official religion is determined by buildings present, or by the governor or faction leader if there are none. Any settlement that has a different official religion than that of the state gets 10% unrest. The faction leaders religious trait also has a huge effect on what religion new characters follow; for example if your faction leader has the Christian trait, nearly all new characters will have the Christian trait as well. For this reason the player should consider religion carefully when deciding who to make the faction heir. If you want to change your state religion from one to another, you have to find a family member with the Religious trait you want, and then make him the heir, then the leader. This can take a long time for the non-Roman factions.
Religion affects a settlement's happiness considerably. Building a Christian church in a mainly Pagan province will result in considerable unrest and possibly even rebellion. Placing a Pagan temple in a Christian town can cause the same problems. Religious unrest can go as high as 100%, and this combined with other factors can make certain provinces impossible to hold. In order to appease their subjects, factions can also decide to build religious buildings of the same religion which is prevalent in a region - even if the faction's state religion is a different one.
Having a governor in a city with a different religion from the official settlement religion causes 5% unrest, as well as a few negative traits, so it is not ideal. Generals can never properly change their Religion trait without modifications to the game.
The Berbers are purely Christian, the Huns and Saxons are purely Pagan, the Sassanids are purely Zoroastrian, all other factions can choose to be Christian or Pagan. Generally speaking, Christianity provides better public order, law, and conversion traits and ancillaries while Paganism provides more military bonuses. Some factions also have certain units linked to religion, such as the Christian Frankish Paladins and the Pagan Lombard Berserkers.
Medieval II: Total War
Empire: Total War
Religion plays a rather minor role in Empire: Total War. Regions that are of a different religion to the faction that controls them will experience happiness penalties due to religious unrest. Factions may choose to build religious centres in regions, which gradually begin to convert the local populace to the faction's religion. Religious centres grant a moderate happiness boost to a region that is predominantly the same religion as the owning faction; however, the happiness bonus is less than that of dedicated entertainment centres. These same centres can also spawn missionaries, who can travel into other regions to begin converting the populace to their religion. This can be used to either cause unrest in other factions' lands or to help stabilize religious unrest in owned regions.
Religion effects diplomacy as well. Factions that share a religion get an increase in relations, while opposing religions gets a decrease. These can easily be overcome however, especially when the religions are similar enough. They make the biggest different when the 2 factions are otherwise neutral.
Some religions are more compatible than others. For example if a Protestant faction rules a region that is entirely Catholic and Orthodox there will be 2 points of unrest, but an entirely Muslim region will have 10 points of unrest. The late-game enlightenment technology "Human Secularism" cuts the effects of both religious unrest and happiness by half, making religion a very minor point in the late-game.
Napoleon: Total War
Religion plays a similar but even smaller role than in Empire: Total War. Regions that are of a different religion to the faction that controls them will experience happiness penalties due to religious unrest, but these penalties are usually small. There is no way to convert regions or to change your own religion, meaning that religious unrest will simply have to be countered in other ways, either through government or happiness buildings, ministers or garrisons.
Additionally there are small diplomatic bonuses between factions of the same religion, and small penalties between factions of different religions, but these bonuses usually are dwarfed by other factors.
Napoleon: Total War: Peninsular Campaign
In the Peninsular Campaign, religion is replaced by political alignment, which can be either Pro-French or Anti-French. The effects of political unrest are much higher than in the base game. Political alignment in a given region can be changed either by Educational Buildings, or by Priests or Provocateurs who are spawned by those buildings. These agents also increase happiness when in friendly cities, as well as unhappiness when in enemy cities.
Additionally, looting a region results in a 50% swing against your factions political alignment in that region.
Shogun II: Total War
Christianity can play a major role in the late-game should factions decide to convert to Christianity in order to access European gunpowder weapons. Christianity will cause severe religious unrest throughout the regions the faction controls due to its unpopularity in feudal Japan.
Christianity allows a faction to access gunpowder options earlier and allows the construction of Nanban Trade Ships, which will absolutely devastate most opposition on the seas. Christian buildings are also better at conversion than Shinto-Buddist buildings and help research the Chi arts faster than their Buddist counterparts. However, conversion hurts the Daimyos honor and causes the faction to have to convert their regions, which is a long process. Since the map is almost entirely Buddist Christianity will slow down expansion. Conversion also causes the faction to lose access to the very powerful Warrior Monk units. Christian factions also get a -40 penalty with Shinto-Buddist factions, although with other Christian factions they get a massive +50 bonus.
The Otomo start as a Christian faction and can never return to Buddism. The Ikko-Ikki have their own religion and can never convert.