Adjacency bonuses аre а new mechanic widely used in Civilization VI to simulate the beneficial or non-beneficial effect terrain and game objects have on each other. It plays into the heavy emphasis this game gives to terrain use and the planning of the new 'urban sprawl' in the form of districts and wonders being built out in the open terrain alongside tile improvements.
What are Adjacency bonuses?
Adjacency bonuses manifest in different ways for different gameplay elements; however, most of the time they manifest in terms of gameplay stats which are added to the element according to its surroundings. For example, 1 point of 
Science yield will be added to each Campus district for each Mountain tile near it.
Adjacency bonuses work for each concerned game element by looking up all six surrounding tiles each turn, and then calculating and conferring bonuses (or penalties) according to what is in these tiles. The bonuses, or penalties, are always in relation to the element itself: it may happen that the same tile with the same stuff on it will affect in 2 or more different ways nearby different elements! For example, a Mine improvement will give a nearby Industrial Zone +1 
Production bonus (0.5 in 
), but at the same time will lower the Appeal rating of all other nearby tiles by -1!
Note that ownership of the bonus-providing features does not matter; what matters are the features themselves! Thus, if the above-mentioned Mine is in the territory of another nation, or City-State, it will provide the bonus nevertheless.
Adjacency bonuses are not static: they may be activated or deactivated at any moment in the game, in response to changing environment. For example, a solitary Farm placed in a plane will have no bonuses; but as soon as 2 other Farms are placed near it (and the Feudalism civic has been researched), it will gain a +1 bonus 
Food! If later you replace one of the farms with something else, the original farm will lose the bonus (because in order to enjoy it, it needs at least 2 adjacent farms). Later, when you research Replaceable Parts the +1 bonus will appear again, due to the remaining one adjacent farm!
Game elements affected by Adjacency bonuses
There are three main element groups where the mechanic manifests:
District adjacency bonuses
The first main domain of Adjacency bonuses are City districts, and more specifically, Specialty districts. Almost all of them benefit from Adjacency bonuses, but each one does so in different ways, according to its specific vocation. District bonuses manifest in the form of raw stat yields, which are provided by the district tile itself (whether it's worked or not) as soon as the district is finished and as long as it remains functional.
Most Adjacency bonuses for districts come from terrain features, such as Mountains, Woods or Rivers - this is why it's very important to plan your district placement as soon as, or even before, you settle a new city. Notable exceptions to this rule are the Industrial Zone, which gets bonuses from improvements (although these arguably also depend on terrain!); and the Theater Square, which gets bonuses from Wonders.
Note that there is no requirement for the features providing the bonuses to be in your territory; for example a Campus built right next to a Mountain which is however in your neighbor's territory will still gain a +1 bonus.
There are three levels of District Adjacency bonuses: Major (+2), Standard (+1) and Minor (+0.5). While the mechanics of the Major and Standard bonuses are self-explanatory, these of the Minor bonus have some tweaks:
- Minor bonuses are rounded down, so a single feature yielding a Minor bonus won't result in any gain at all (since 0.5 rounds down to 0); for example if a Campus is adjacent to a Mountain (which yields Standard bonus) and one District (yielding Minor bonus), it will get a total of +1 Bonus, not +2.
- If a specific District gains Minor bonuses from more than one feature, note that each individual feature's bonus is counted separately, and again rounded down! So, if an Industrial Zone is next to one Mine (which yields a Minor bonus), and one Lumber Mill (again, yielding a Minor bonus), it will gain 0 Adjacency bonus; if it's next to two Mines, one Lumber Mill and one District (which also yield Minor bonus), it will gain +1 bonus (not +2!).
Each District, including the City Center and 'engineering' Districts (such as the Aqueduct), confers a Minor Adjacency bonus to all nearby districts. This is why it is sometimes beneficial, in the absence of better options, to cluster all districts in a city next to each other. In Rise and Fall the Government Plaza district confers a Standard adjacency bonus to all Specialty districts, which is in addition to the Minor one! It's thus a good strategy to build it in a strategic location where you can later surround it with other Districts which make use of its additional bonuses. Theater Square districts benefit greatly from a Government Plaza, because their only other Adjacency bonus comes from Wonders, which are obviously hard to obtain.
Tile improvements adjacency bonuses
The second domain of Adjacency bonuses are Tile Improvements. The Farm's technological development over time is based solely on adjacency bonuses with other Farms, and the more adjacent farms there are, the better! This incentives players to build extensive areas of flatlands (and later, Hills as well) as Farms, simulating real life.
Most Special and Unique Tile improvements also provide, or benefit from some type of Adjacency bonuses. Most of these are based on terrain features, resources, or even other improvements. Sometimes, even the placement itself is conditioned to a specific feature or resource (for example, France's Chateau has to be near a River, while Indonesia's Kampung has to be near a sea resource). In some other cases, however, the improvement benefiting from the bonus cannot be placed next to another improvement of the same kind; this is the case with the Mekewap. These diverse conditions force the player to re-evaluate his placement strategies with each different civilization, making each game unique! Each different situation will also play into the individual civilization's gameplay style, sometimes incentivizing creative planning and combining an improvement with other game elements, other times incentivizing the formation of large conglomerates of same-type improvements (as in the case of Australia's Outback Station).
Note that, just as in District adjacency bonuses, bonus-providing features or game objects don't need to be all in your territory in order to work.
Appeal
This special feature of land terrain is formed entirely by Adjacency bonuses. Surrounding tiles determine how 'appealing' or 'disgusting' each particular tile is, according to what terrain or features are found on them. Unlike the other two domains, which are well-explained in the game and in individual game object descriptions, Appeal mechanics are often overlooked, though relatively simple. And, they are largely controllable by the player and his placement of (or sometimes removal of) other game features around his cities.