Beliefs are the practical expressions of the divine powers. Similarly to Pantheons, they take the form of gameplay bonuses and can increase yields of particular gameplay elements (buildings, districts, etc.), allow access to new ones, or modify game mechanics. However, unlike Pantheons, they work only in the cities which currently follow their particular Religion, even inside the empire of the Religion's founder.
There are four main types of belief in the game: Follower Belief, Worship Belief, Founder Belief, and Enhancer Belief. Every Religion can only have a maximum of four beliefs, one of each type.
Each Religion is attributed with beliefs, which are in fact gameplay bonuses.
Founder Belief - Only applies to the founder of that Religion, and to all Kongolese cities if this Religion has established itself in the majority of the cities.
Follower Belief - Provides bonuses for any city following this Religion, regardless of civilizations.
Worship Belief - Grants a specific Tier 3 worship building that can be constructed in the Holy Site. Each building provides different bonuses, but again they work for any civilization whose city adopts this particular Religion. All worship buildings can be bought with Faith. Only one worship building can be built per Holy Site and cannot be changed even when that Religion is no longer dominant in that city.
Enhancer Belief - Designed to help the spread and defense of the Religion, only applies to the founder of that Religion, and to all Kongolese cities if this Religion has established itself in the majority of the cities.
When you found your Religion, it starts with two beliefs. Of these two, the first one to choose is a mandatory Follower Belief, and the other may be a Founder Belief, a Worship Belief or an Enhancer Belief. Later, you may add two additional beliefs to your Religion using an Apostle'sEvangelize Belief action. This will destroy the Apostle, so you'll need to sacrifice two Apostles to have the maximum number of beliefs allowed. You may add these beliefs whenever you wish - that is, there is no time limit to this possibility, and the limited number of Religions that can be created ensures that you will always have some beliefs to choose from.
All beliefs are unique, chosen from a common pool, and can be assigned to only one Religion. That means that later-founded Religions cannot choose beliefs which have already been assigned to earlier Religions. Thus, the more Religions are founded, the fewer beliefs there will be to choose from and if you are late at founding your Religion, your choice will be more limited.
These benefit any city that follows the religion, regardless of who its Founder was. Players who have not Founded a Religion will mainly benefit from this type of Beliefs.
Belief
Effect
Choral Music
Shrines and Temples provide Culture equal to their intrinsic Faith output.
These allow any city that follows the religion to build (or use Faith to purchase) Tier 3 worship buildings in its Holy Site districts. Note that each Holy Site may contain only one worship building, which stays there permanently. If the dominant religion in the city changes later, and the new one has a different Worship Belief, you will be able to replace the worship building that already exists there.
These benefit only the civilization that founded the religion (or the Kongolese thanks to Mvemba a Nzinga's leader ability). Their starting benefits may be slightly weaker than other belief types, but they will work for each converted city or for each follower, potentially increasing the benefits far beyond other belief types.
These beliefs aid the spread and defense of the religion. Only the founder of the Religion will benefit from these (or the Kongolese thanks to Mvemba a Nzinga's leader ability).
Incredible belief, though it does not sound like it. Culture is an important yield no matter what Victory conditions and what civilizations you are playing. Notable civilizations who love this belief are ones that need a lot of early game Culture to hit an important timing: Indonesia, Vietnam, Poland, or any civilizations who love building Holy Sites.
Divine Inspiration
Civilizations that like building wonders can particularly benefit from this belief. Try to get this one if playing as Qin Shi Huang, Egypt, or France.
Feed the World
This belief is a lot better in Gathering Storm, as its bonus Food and Housing can allow cities to grow quickly. Civilizations with difficulty feeding their cities, like Russia or Canada, can take advantage of this belief.
Jesuit Education
This belief is best used in the hands of a civilization with great Faith generation, like Ethiopia, or a civilization with some religious incentives but try for a Cultural or Scientific Victory, like Arabia or Indonesia.
Religious Community
Never pick this belief if playing in a version before Gathering Storm. In Gathering Storm, it does not matter which civilization you are playing as, but who your neighbor is. Pick if your neighbors love building Holy Sites and developing them, most notably Arabia and Khmer.
Reliquaries
If you are intending to playing a Cultural game with Relics and the Mont St. Michel, pick this one.
Warrior Monks
Another belief that is better in the expansions. In the base game, this belief should rarely be picked, as detailed here. On the other hand, Gathering Storm combines Burial Grounds with this belief, making it slightly better, especially for Poland, but only if they are playing a Religious/Domination game, and not a Cultural one.
Work Ethic
An underwhelming belief before Gathering Storm. On the other hand, Gathering Storm changes it to be much better for civilizations such as Japan, Russia, Mali, and Brazil, which can easily get massive adjacency bonuses.
Zen Meditation
One Amenity never solves anything, especially considering this is the only belief that comes with a requirement. There is always something else better.
This worship building is most useful for a player going for a Cultural victory. It can augment a Holy Site's Tourism output with its slot for a Great Work of Religious Art. However, the restriction on Great Work type makes this highly dependent on which Great Artists can be obtained. Only 6 Great Artists producing only 12 Great Works of Religious Art are available, and even these are better spent on theming some Art Museums. (Read more here.) Overall, this is a pretty bad and unreliable choice for your Worship belief, even for a cultural civilization, since you cannot control if you will have Great Works of Religious Art or not. If you insist on using your Holy Site to produce Tourism, it is usually better to focus on generating some Relics to place in your Temple instead and pick a better worship building. Relics are much more powerful in the long term, especially if you have special benefits from them (Jadwiga, a religion with Reliquaries) or can acquire them more easily (Khmer).
This worship building provides increasing amounts of Faith with the passing of time, making it advantageous to build or purchase early on. However, the Dar-e Mehr should almost never get picked up, considering that Religious Victory has a really narrow window of opportunity, rendering it counter-synergistic with the one purpose it is built for. Not to mention, if this building gets pillaged, rebuilding it means nullifying all the extra Faith you accumulate over the eras, and there is likely to not have much time left for you to build this Faith up again. This makes it an underwhelming choice, but somewhat better for Cultural civilizations, since they will usually avoid war and it can generate enough faith to buy expensive Naturalists and Rock Bands by the endgame.
This worship building allows the Holy Site to increase city growth, especially if combined with the Feed the World follower belief. Note that the Food bonus does not do much good without corresponding Housing capacity, so this Worship building is helpful either when your cities are settled in Plains, Tundra or Desert where Food sources are scarce, or when you are playing as a civilization with early to mid-game Housing bonuses, like India. Otherwise, all this building does is make you run into Housing issues even sooner, which is not advisable.
In Gathering Storm, this building now also grants 1 extra Housing, which makes it a bit stronger for any civilization that loves building populous cities.
This worship building also boosts city Production, which helps in all instances. Choose it if you don't have any particular strategy for now, or if another Worship Belief you wanted is already taken.
It is especially useful if your cities are in areas with lots of grassland tiles (which give no Production on their own) or rainforest (which, outside of Gathering Storm, cannot receive Lumber Mill improvements). This is undoubtedly one of the best Worship buildings, so the AIs place a special emphasis on it. If this is what you want as your Worship building, you should get it right when you found your Religion, don't wait until you evangelize it.
The Mosque can help speed up the spread of your religion, making it very useful when pursuing a Religious Victory. The main advantage of it is that you only really need one. That makes it a much cheaper investment than all the other religious buildings (which are frankly expensive for what you get), though later in the game you may want to dot a few more around the globe.
This building has extra synergy with the Hagia Sophia and the Oratorpromotion, as it can give Apostles an unprecedented seven Spread Religion charges.
Similarly to the Gurdwara, the Pagoda also helps city growth, this time by providing additional Housing. Bearing in mind how often Housing restrictions turn into a problem in this game, we can say that this bonus is even better than the Gurdwara bonus.
In Gathering Storm, the Housing bonus is now transferred to the Gurdwara, the Pagoda now grants 1 Diplomatic Favor per turn instead. This will help any civilization that wants to go for a Diplomatic Victory, or just simply wants a stronger standing in the World Congress.
This worship building provides an Amenity, allowing cities to grow faster and be more productive. Because of this, it is invaluable to warmongering civilizations who may need some more Amenities due to war weariness. However, if you have a lot of Luxuries, or the Colosseum, then the extra Amenity just becomes redundant.
This worship building simply boosts a Holy Site's Faith output, making it easier to purchase units, buildings, or Great People. It doesn't have an additional effect, though, so you should really think about whether you could really use tons of Faith in your strategy, or if you'll need something extra.
This worship building allows the Holy Site to boost your civilization's scientific research - something quite strange for a worship building. It may prove very useful in all situations, though, especially if you don't have any other particular strategy regarding worship buildings, and if you're not pursuing a Religious victory. Arabia will want to take this building if they can, because they are strong in both religion and science. You may also want to take it for some extra Science output if you plan to focus on maximizing other yields.
The amount of Gold you get back from this is almost never worth the amount of Faith expended to convert other cities. It was removed in Gathering Storm, but before that, Tithe is the better choice for Gold.
Cross-Cultural Dialogue
This belief really comes into play in the late game, where it can bring you massive amounts of Science. Arabia can use this belief incredibly well, because of their tendencies towards Scientific and Religious Victories.
Lay Ministry
This belief relies on what your enemies do more than what you do, and the amount of yields you get back are not worth the Faith you spend to convert those cities.
Papal Primacy
Decent in vanilla, not worth picking up since Rise and Fall unless you are playing as Tamar.
Pilgrimage
A so-so belief for ones seeking a Religious Victory. The amount of Faith rewarded after converting a city somewhat helps shoulder the increasing costs of religious units.
Religious Unity
Free Envoys make this belief a good choice for any civilization with a love for city-state diplomacy, such as leaders like Pericles and Matthias Corvinus, if they somehow can fit a religion in their strategy. Tamar should make this belief a priority, because she will already be motivated to convert city-states to her religion, due to her leader ability.
Sacred Places
It is most reliable when you are the one who prioritizes building Wonders and not others. Best for Qin Shi Huang, Egypt and France.
Stewardship
Similarly to Lay Ministry, very unreliable.
Tithe
A fairly strong way to make Gold off of your religion, especially in Gathering Storm, where it is buffed. Best for civilizations that are otherwise without any unique way to earn significant amounts of Gold, especially those paying high maintenance costs for large armies, such as Scythia and Byzantium.
World Church
Similar to Cross-Cultural Dialogue, but for Culture. Civilizations such as Russia, Poland, and Khmer can use this belief well, as they all have dual Religious-Cultural tendencies.
A great choice for any militaristic civilization, especially if they also have a religious skew. Spain, Byzantium, and Poland should all aim for this belief. Despite not having a religious skew, Babylon can potentially found an early religion and pick this belief to help them on their path to a Domination Victory.
Defender of the Faith
This belief is a good choice for any defensive civilization or ones that anticipate an attack from aggressive neighbors.
Holy Order
Faith conservation is useful under any circumstances, except for civilizations with a lot of Faith. A solid belief choice if you intend upon trying for a Religious Victory.
Holy Waters
Can be useful in a game where you seek a Religious Victory but are playing against a lot of other religious civilizations. Otherwise, this is a wonky belief that is very hard to successfully pull off.
Itinerant Preachers
The bonus provided by this belief is far too weak to be of any consequence, and it should usually be passed over.
Missionary Zeal
This belief allows your religious units to reach civilizations far away in less time. Don't pick this belief on a water map, since it does not make embarking and disembarking easier, or shorten the time required to cross oceans.
Monastic Isolation
This belief can work as a backup for the Khmer or anyone who builds Mont St. Michel, if they care about establishing their religious foothold, as their Missionaries and Apostles are regularly killed in theological combat. Most of the time, however, they collect Relics to pursue a Cultural Victory, which renders this belief useless.
Religious Colonization
This belief is a lot more useful than it sounds. Normally, civilizations that have been converted to your religion can easily become unconverted if they found enough new cities. By picking this belief, once your religion has taken ahold of a civilization, they will have to take actual steps via Inquisitors in order to remove it.
Scripture
Passive religious pressure is very weak, and this bonus does little to improve it, so there are better beliefs to pick.
Culture equal to their intrinsic 
Faith output.
Food equal to their intrinsic 
Housing.
Tourism yields from 
Relics.
Production for each follower.
Amenity in cities with 2 specialty districts.
religious art.
Science
Gold for each city following this Religion.
Envoy at each city-state when it first adopts this religion, in addition to any envoy from a city-state quest.
Combat Strength near foreign cities that follow this Religion.
