Barbuda is played by four people (no more, no less) with a normal deck of 52 cards. There are 7 different "games" in Barbuda, 6 of which are variations on normal card games such as hearts, spades, etc. These games are: Kings and Jacks, Queens, Hearts, Nil, Last Two, Tricks, and Fantan. Except for Fantan, each of these games works in a manner similar to normal hearts: each player is dealt 13 cards, each player must follow the suit led (if possible), the highest card in the suit led wins the trick, etc. The only difference is the scoring and the fact that the dealer always leads first (and no cards are passed to other players or anything like that).
The game starts by cutting for positions, with the highest card being the first dealer, the second highest card sitting to their left, the third highest sitting across from them, and the lowest card sitting to their right. The dealer deals the first hand, then decides which of the seven games they want to play (after looking at their own cards). After playing that game once, the player on the left becomes the dealer and gets to select the next game (after dealing, of course). This continues, with the deal always going to the left, until each player has dealt 7 hands (for a total of 28). Each player is only allowed to call a given game once, so eventually each player will call each game once (and will have no choice on their last call). After all 28 hands have been played, each player totals up their score to see who has won.
Kings and Jacks: In Kings and Jacks, each king or black jack taken by a player lowers their score by 5 points, while each red jack taken lowers their score by 2 points. This game is often very dangerous, since a total of 34 points can be lost on very few cards.
Queens: In Queens, each queen taken by a player lowers their score by 6 points. While only 24 points can be lost in this game, all the points are concentrated in just four cards.
Hearts: In Hearts, each heart taken by a player lowers their score by 2 points. Note that the queen of spades doesn't matter, cards are not passed to other players, "running" doesn't count, and the player with the two of clubs does not automatically lead (the dealer always leads). However, hearts must still be "broken" before any player is allowed to lead hearts (unless they have no other suits left). This game is generally considered fairly safe since the points are spread out and only 26 points can be lost.
Nil: In Nil, each trick taken by a player lowers their score by 2 points, regardless of the cards in the trick. This game is generally considered to be the safest of all games since there are only 26 points to be lost and it is almost impossible to take them all.
Last Two: In Last Two, taking the second to last trick lowers a player's score by 10 points, while taking the last trick lowers a player's score by 20 points. This game is considered extremely dangerous, since 30 points can be lost and they will always be lost on only two tricks.
Tricks: In Tricks, each trick taken by a player increases their score by 5 points. In addition, the dealer can call a trump suit if they wish, which means cards of that suit always win over cards of other suits (players must still follow suit if they can, though). Not calling a trump suit is called "no-trump". While a player can only gain points in this game, it is important to get at least five to seven tricks, since there are 65 points available.
Fantan: In Fantan, everything works differently. Instead of playing cards like the other six games, the dealer calls a card number from 2 to an ace (aces are always high in Barbuda). The dealer then must play a card of that number (but of any suit). Then each player in turn must play a card of that same number, or a card one above or one below a card that has already been played (of the same suit). If a given player has no legal cards they can play, they just say "pass" and skips their turn. A player cannot skip their turn if they have even a single legal card they can play. Fantan is scored by the order in which players get rid of all the cards in their hands. The first player to get rid of all their cards increases their score by 50 points. The second player to get rid of all their cards increases their score by 25 points. The third player to get rid of all their cards increases their score by 10 points. The last player decreases their score by 10 points. Obviously, Fantan is one of the most important games, since so many points are at stake.
Barbuda is always played for money, with the stakes being anywhere from a penny a point to a dollar a point or more. Since the scores of the four players will always add up to zero (and if they don't you've made a scoring error), the players with negative scores pay the players with positive scores. Note that scores are only guaranteed to add up to zero if all 28 hands are played, so no one is allowed to leave the game mid-way through.
After the dealer of a given hand has called the game (and the suit or number, if necessary), the player to their left can declare that they are "doubling" the dealer. This means that, in addition to the normal scoring, the player gets a number of points equal to the difference between their final score for the hand and the dealer's final score. These points are taken from the dealer (whose score is reduced by the same amount). However, if the player gets fewer points than the dealer, the dealer increases their score and the player decreases their score by the difference. After the first player has declared whether they are going to double or not, the next player must make the same decision, followed by the third player. Players cannot change their minds after they have declared whether or not they are going to double. The dealer cannot decline any doubles and non-dealers cannot be doubled by the dealer (or by anyone else). However, the dealer can "re-double" if they desire (after all doubles have been made). This means any points gained or lost because of doubling are themselves doubled (the dealer cannot selectively re-double a given player--it's all or nothing).
The standard way to make a scoresheet for Barbuda is to make six columns. The first four columns are topped by the player's names (in order, starting with the first dealer). These columns are used to keep track of each player's current score, with each line denoting a single hand. The fifth column is used to keep track of the current total of all the player's scores. This column should be filled in the moment the game is called (adjust the total by -34 for Kings and Jacks, -24 for Queens, -26 for Hearts or Nil, -30 for Last Two, +65 for Tricks, and +75 for Fantan). After the hand is finished, add up all the players' scores to make sure the total matches this column (which is called the "check" column). The sixth column should be used to mark the game called (KJ, Q, H, N, L2, T, or F).
An additional four columns, one for each player, should be used to keep track of any points gained or lost from doubling. Keeping track of doubling separately makes it easier to correct scoring mistakes and makes it clearer what happen during each hand. The scorer may also want to make a grid of players vs. each of the seven games so that each game called by a given player can be marked. While not essential, this makes it easier to see which games a given player has left to call.
The only variants of Barbuda that I know of are changes to the scoring for Fantan. Some players don't like how important Fantan is, so they change the scoring to 35 for first, 25 for second, 15 for third, and 0 for last. This reduces the difference between first and last to 35 points instead of 60.