TL;DR

Rating vs. Scoring

Rather than rating changes happening directly as a result of final points standing, games are scored by giving each player a number of wins or losses.
The amount of rating points you gain or lose when receiving wins or losses will vary, depending on what your rating is currently, how consistently you maintain your rating, and the average rating of players in the game.

4-Player FFA

In games where the average rating is low, or close to 1500, 1st place will be awarded 3 wins, 2nd place gets 1 win, 3rd place gets 1 loss, and 4th place gets 3 losses.
In games where the average rating is high, 1st place will be awarded 4 wins, and each of the remaining players will be penalised 1.33 losses.

Teams

A team's average rating is calculated in favour of the higher-rated player, at a weight of 2/3 to 1/3 from the lower-rated.
Both players on the winning team receive 1 win, as calculated against the average rating of the other team. Both players on the losing team, 1 loss.

Glicko-2

Variant ratings on chess.com use the Glicko-2 rating calculation system. This system of scoring was specifically developed for multi-player games, and represents a mathematically fair way of assessing point gains and losses dependent on player's skill, consistency, and the elements of luck which arise from any multi-player game.

The heart of Glicko-2 is the value of Rating Deviation, or RD. Ratings are dependant on a probability function, as opposed to a single, defined value. When you see your rating on chess.com/variants, that value is the system's best approximation of your skill, to be interpreted as the expected value, but surrounded by a bell-curve distribution, with a standard deviation given by your RD:

Players with a lower rating deviation have a more predictable rating, meaning the system has a larger degree of confidence that their ratings are close to the given values. These players' ratings consistently stay within a narrow window above and below the reported rating. As such, the system doesn't change their ratings as dramatically. If two equally-rated players each lose to a field of equally-averaged opponents, the player with the higher rating deviation will lose more points than the other. Likewise for points gained after a win. The system

Anyone looking for an in-depth explanation on the specifics of Glicko-2 can find more information or open source implementations on the creator's website

4-Player Scoring (FFA)

The Glicko-2 system is responsible for changing ratings after wins and losses, but the other component of scoring is how many wins/losses are assessed in each game. In the past, there have been many in-depth discussions about how exactly to reward/penalise players in 1st, 2nd, 3rd, and 4th place after a 4-player game. From those discussions and prior tests, the system for chess.com/variants has been implemented as follows:

At lower levels of play (beginners), the 1st place finisher will be awarded 3 wins, 2nd place will earn 1 win, 3rd place will be penalised 1 loss, and 4th place, 3 losses. This will be the case for all FFA 4-player gamemodes where the average rating of all 4 players is close to 1500.

At higher levels of play, players are expected to be more competitive, and not simply take the easy route out when as the game comes to its conclusion. Therefore, it has been decided that for games with higher average ratings, 2nd place should not be awarded points, but rather penalised for not winning outright. This encourages a cooperative spirit among players who aren't leading in points to fight every man for himself, and negotiate to the top of the standings at every opportunity. In these higher-rated games, 1st place will also gain a larger reward of 4 wins, while 2nd, 3rd and 4th place will each be penalised equally with 1.33 losses, thus adding to the notion of fight as equals to prevent the common enemy from winning. The mentality in these games should be firstly, to equalise the playing field, but secondly to gain marginal advantages until you cannot be stopped, even with the eventual 3-vs.-1 cooperation of your opponents. You will gain nothing by fighting against the 3rd or 4th place finishers only to land in 2nd behind the winner.

The determination of the boundary between higher-rated and lower-rated will be set by the admins, and may be a different value for different modes of play. Wherever that boundary is set, the transition from +3/+1/-1/-3 to +4/-1.33/-1.33/-1.33 will be a gradual one modelled by a series of logistic functions, dependent on the game's average rating, and where the admins have set the Solo Point.

Currently, the solo point is set at the following values for these popular gamemodes:

2500: Standard 4PC, Arena of Legends, The Four Horsemen
2200: War for Throne, Diamond Wall, Glass Labyrinth
2000: Collateral Worlds, Arena of Grasshoppers, Battle of Castles
1800: King Safety

*These determinations may be adjusted as the distribution of ratings within these classes rise or fall.

Exceptions

The following exceptions can cause scoring changes to occur in ways contrary to what was explained above:

Pure FFA scoring

In these gamemodes, scoring will always be handled as if the average rating was 1500, i.e. +3/+1/-1/-3. Notable implementations of this scoring system apply to Chaturaji, Labyrinth 82, and 4P Giveaway.

Pure Solo scoring

Similar to pure FFA, this scoring system always acts as if the average rating is at the solo point. 1st place gets 4 wins, and 2nd-4th all lose equally with 1.33 losses. Games such as Unison use this scoring system.

Points scoring

Just like the other modes, there are still 4 wins and 4 losses to be distributed. However, with this method, the wins and losses are distributed proportionally to the amount of points each player earnt in the game, and how far above or below the average it stands. An example of how the wins and losses would be distributed in a game with points scoring is shown below:

3-Player scoring

Is the same as 4-player for most modes, except there are 3 wins and 3 losses to be distributed. The FFA scoring differs slightly in that 2nd place will receive both wins and losses determined by the logistic function morphing from 0.75 wins+0.75 losses to 0 wins+1.5 losses. The breakdown of this morph is shown below, where the red and blue lines are both assessed upon 2nd place:

2-Player scoring

1 win, 1 loss. Shouldn't be that difficult to figure out who gets what.

Ties

If 2 or more players finish a game with the same amount of points, the rewards/penalties between their placements will be divided evenly and applied for each. For example, if two players are tied for 2nd and 3rd in a game where the scoring was +3.38/+0.14/-1.15/-2.37, then each of them will receive 0.07 wins, and 0.58 losses.

Aborts

If a player disconnects or resigns near the beginning of a game, it can result in a horrifically unbalanced game for the remaining players. In these instances, the game is aborted – or cancelled – so that the players who didn't choose to abandon the game can get off to a more equitable start. Aborting games intentionally to play for a strategic advantage is against the spirit and rules of the game and is the height of unsportsmanlike play. To actively discourage this practice, point penalties (and on occasion, playbans) are assessed to players who abort games. Aborting when your rating is low, or infrequently enough that it can be attributed to unintentionality will only give minor penalties, but for players who abort frequently, or have high-enough ratings to know better, those penalties can be quite steep.

Teams Scoring

In teams games, there's nothing fancy with the number of wins and losses. The players on the winning team get the win, and those who lose get a loss. The only complication is how the Glicko-2 system works to change ratings with the information of 1 win, or 1 loss. In an FFA game of 4- or 3-player, the wins and losses are computed against a hypothetical player with a rating equal to that of the game average. This allows players with ratings lower than the average to be rewarded more favourably when beating out a stronger player pool, and penalised a lesser amount when receiving losses against players stronger than they are, as would logically make sense.

For teams, instead of giving wins or losses versus the average game rating, players' wins and losses count as if competing against a player with the average rating of just their opposing team's players. This happens with the caveat of that average being weighted in favour of the stronger (higher-rated) player. Because players on a team can communicate with arrows, chat, or through external means, the strength of a team can be heavily or almost solely reliant on the player with a better understanding of the game. As such, that player's average will count twice as much as the player with the lower rating. I.e., 2/3 versus 1/3. For example, if a team comprised a player rated 2154 and another rated 2436, the members of the opposing team would each receive either 1 win or 1 loss versus a player with a rating of 2342.

Other Notes and Resources

A natural consequence of how games work within the scoring framework is that once a 4-player game has been reduced to 2 players, the outcome, scoring-wise is often already decided. For that reason, players can resign to win (claiming the win), be forced to claim a win (autoclaim), or be forcibly resigned (autoresign). If that sounds complicated or scary, please find a more detailed explanation here.
For transparency and a fairer game experience for all, all 4-player games will broadcast a message at the top of that game's chat box breaking down the scoring for that particular game. For all rated non-teams modes except for the points-based scoring, that message will look something like the examples shown below (points games will have a concise standardised explanation of how scoring will work in that game):
Theory is a completely different discussion than scoring, although a thorough understanding of the scoring system definitely will give you an advantage in your gameplay, and change the way you approach theory. One of the first steps in that journey is understanding that even in solo-esque play, cooperating with another player or players will give you an advantage over those that are playing with an army of 1. There is a distinct difference between cooperation (allowed and encouraged) and teaming (prohibited), which has been the subject of many write-ups prior to now. If this is a new idea for you, please seek those explanations on your own, but start here and here.
The article above attempts to explain how rating changes occur, but the categories of different ratings, and separation of rating types have been explained elsewhere, so were omitted here. Those descriptions can be found here.
Some players may wonder about how the admin team has chosen to assign scoring types to different positions and gamemodes. Although providing that reasoning would be difficult for each and every game, there is an overarching way in which the custom games team evaluates positions, and finds a scoring scheme that fits their ideas. An introduction to this evaluation process is described here.
With any questions about what was described here, or for a general Q&A regarding specific implementations on the variants server, please leave your comments below!

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