Source: http://www.quoitsdirect.com/QuoitRules.htm

STARTING A GAME:
The quoit boards are placed 18 feet on center (from hub to hub).
One-on-one - You pitch from the same side of the boards when you travel to the opposing board. Non-diagonally. (One-on-One Positions).
Two-on-two (a.k.a. couples or doubles) - Your partner pitches from the diagonal side of the board (Two-on-Two Positions).
There are 4 quoits (2 for you and 2 for your partner). You can determine who throws the first pitch of the first game. After the first game the loser of the prior game makes the first pitch to start the new game.
Pitching turns alternate with each quoit.
The pitcher's forward foot may not extend beyond the hub of his "home" board.

DETERMINING THE SCORE:
The first person or team to reach a score of 21 wins the game.
Every "ringer" (meaning the quoit landed on the hub) is worth 3 points.
If a player should make a ringer and have his second quoit closer to the hub than his opponent, he receives 4 points (Three points for the ringer and 1 point for the quoit closest to the hub).
Each ringer pitched counts as 3 points, except when the same player pitches two ringers, one directly on top of the other. This player is then awarded 3 additional points, for a total of 6 points.
When the first player pitches a ringer which is topped by an opponent, it is the opponent only who receives 3 points.
When the first player pitches a ringer which is topped by an opponent and then topped by the first player again, the last ringer made by the first player counts as 3 points for the first player. As a premium, when four ringers are made, the player pitching the last ringer wins the game.
Every "leaner" (meaning the quoit is leaning against or on top of the hub, but not a ringer) is worth 1 point.
If there are no ringers and leaners the person who throws the closest quoit gets one point. If the same person's second quoit is the next closest he/she gets two points.
The rings on the board are used to determine the closest quoit. If you can not determine the closest quoit from the front side (closest side of the quoit to the hub) you can usually determine it by comparing the backside (the side of the quoits furthest from the hub) to the outer rings.
If a quoit is touching the ground, or has touched the ground, it is a "dead quoit". Remove it from the board before the next pitch. If you don't remove it any quoits that touch a dead quoit are also dead.
A quoit can be hanging off the side of the board. As long as it has not touched the ground the quoit is still good.
This is called the daylight rule because you can see "daylight" through the quoit since it is hanging off of the board. This rule is sometimes disputed so decide if you want to play "Daylight Counts" before you start playing against your opponent.
Interference due to a wandering dog or small child (which sometimes happens) is grounds for a "re-throw" (a.k.a. do-over) if the quoit is deflected during release or in mid-air.

Source: https://www.tradgames.org.uk/games/Quoits.htm

The Northern Game
The Association of Amateur Quoits Clubs for the North of England was the name give to the official body that laid down the 15 rules that were published in "The Field" in 1881. This set of rules constituted what is now called The Northern Game and it has remained largely unchanged ever since. Although the Northern Game has lost some of it's popularity, it is still played seriously and enthusiastically in the North of England under the auspices of The National Quoits Association which was formed in 1986 and is undergoing something of a resurgence of late.

The Long Game
Quoits however, is a split sport, much like the game of Rugby, because independently, Wales and Scotland have long been playing a distinctly different game with a similar proud history. In fact, "The Long Game" or "The Old Game" is also played in parts of England, notably East Anglia and Liverpool , but is far more rare in that country as the new century begins.
In the Northern game, if more than one quoit lands on the hob, then only the top quoit scores whereas in the Long Game, the scoring is simply that a player scores a point for each quoit nearer to the pin than his opponent. The reason for this is that the pin in the Long Game is knocked down flush with the clay surface so encircling the pin is not really a significant part of the game. The Long Game then is more akin to Bowls in play with the pin acting as a simple target. However, this simplicity belies the difficulty of the game which, in common with many celtic sports, is a test of strength as much as skill. The hobs are 18 yards apart, fully 7 yards greater than the Northern Game while the quoits are typically around 9 inches in diameter and weigh up to 11 pounds, almost double that of the Northern game.

East Anglian Quoits
The long game is still played in England at the turn of the century but in a modified form that sees the quoits reduced in size and somewhat different rules being adopted. The English stopped fielding a side at the internationals some time ago presumably because the differences between the two variations had grown two large and the gap could not be easily bridged. The East Anglian game is unique in that ringers score a clean two points regardless of the opponent's efforts and are immediately removed prior to the next throw. Quoits on their backs and quoits that land inclined in a backwards direction are discounted and also removed immediately. The quoits tend to be smaller than those used in the Scottish and Welsh games but the distance is maintained at 18 yards.

Sward or Lawn Quoits
The difficulty of maintaining the clay squares and the muddiness that can occur on a wet day makes genuine traditional quoits a rare sport. However, all is not lost because for many decades people have been playing a more simple version by simply putting 2 stakes in a lawn. Organisers of fetes, horse fairs and similar public functions sometimes set up temporary quoits pitches in this way and such games are often referred to as Sward Quoits. Sward Quoits has it's own varying complexities depending upon the type of ground being played upon and the height of the hobs and so forth. People wishing to play quoits in their back gardens or any open space happily emulate Sward Quoits in the same way.