Charlemagne is a trick-taking game played in Eastern Canada and across the border in Maine. It is clear from the card ranking that it is based on Euchre, and it could be considered as a version of Bid Euchre. In Canada it is played with two Jokers as top trumps, ranking above the trump Jacks. In the US rules that I have seen, there are no Jokers and instead the trump 9 is the top trump. The Canadian form of the game will be described first.
There are four players, two against two, in fixed partnership, with partners sitting opposite each other. The deal, bidding and play are clockwise.
A 34-card pack is used, consisting of cards from Ace down to Seven in all four suits, plus two distinguishable Jokers. If one Joker is coloured and the other not, then traditionally the uncoloured Joker is the top trump and the coloured Joker is second. If the Jokers look the same, they must be marked to show which is bigger.
In trumps, the Jokers are highest, followed by the Jack of trumps (Big Jack), then the Jack of the other suit that is the same colour as trumps (Little Jack), then A, K, Q, J, 10, 9, 8, 7. In French, the Jack of trumps is called "le bar" and the Little Jack is "le petit bar". The word "bar" is clearly adapted from "bower", the term for the trump Jacks in Euchre, which is in turn derived from the German word "Bauer", which also means a peasant or a playing-card Jack.
In non-trump suits the cards rank from high to low A, K, Q, (J,) 10, 9, 8, 7. The suit that is the same colour as the trump suit has only 7 cards, omitting the Jack, which is transferred to the trump suit.
Example. When hearts are trump the suits rank as follows, from high to low.
J, 
J, A, K, Q, 10, 9, 8, 7A, K, Q, 10, 9, 8, 7
A, K, Q, J, 10, 9, 8, 7
A, K, Q, J, 10, 9, 8, 7 Any player may deal first, and the turn to deal passes to the left after each hand.
The dealer shuffles, the player to dealer's right cuts and the dealer deals eight cards to each player. The method of dealing varies - for example the cards could be dealt two at a time. At some point during the deal the dealer places two cards face down in the middle of the table to form the kitty.
There is a single round of bidding, beginning with the player to dealer's left and ending with the dealer. Each player in turn either passes or names a bid, which must be higher than the previous bid if any. The possible bids in ascending order and their meanings are as follows:
If the bid was 5, 6, 7 or 8, the bidder picks up the two kitty cards without showing them to make a hand of 10 cards, discards any two unwanted cards face down, and names the trump suit. The player to the left of the bidder then leads to the first trick.
If the bid was Mulot there are no trumps except the two Jokers, and all the suits rank from high to low A-K-Q-J-10-9-8-7. The kitty is not touched. The bidder's partner discards his or her cards face down and takes no part in the play. The bidder leads to the first trick.
The procedure for Gros Mulot is the same as for Mulot except that at the end of the first trick the bidder's hand is spread face up on the table so that everyone can see the bidder's cards. Play then continues with the bidder playing cards in turn from the table.
If the bid was Charlemagne, the bidder picks up the kitty and the bidder's partner pass the bidder one card. The card is passed face down and without discussion between the players. The bidder then discards 3 cards, names trump and leads to the first trick. The bidder's partner discards his or her cards face down and takes no part in the play.
The procedure for Gros Charlemagne is the same as for Charlemagne except that the bidder either takes the kitty or asks partner for a card, and then discards 2 cards or 1 card respectively before naming the trump suit ands leading to the first trick.
Players must follow suit if possible, and for this purpose the Little Jack belongs to the trump suit, not to the suit printed on it, as do the jokers. A player who cannot follow suit may play any card. The highest trump wins the trick. If no trumps are played to the trick the highest card of the suit led wins it. The winner of the trick leads to the next.
In Mulot and Gros Mulot, the rules of play for Jokers are different. A player who has a Joker must play it if unable to follow suit. If a Joker is led, the holder of the other Joker must play it, and anyone who does not have a Joker is free to play any card. Because the bidder leads to the first trick in Mulot, it can be bid holding the coloured Joker, which the bidder will lead, forcing the holder of the other Joker to take the trick (but failing if the other Joker was in the kitty).
A cumulative score is kept for each team. The score depends only on whether the bid was successful or not. If the bid succeeds, the bidding team adds the appropriate number of points to their score; if it fails the opposing team gets points. The scores are as follows.
The first team that achieves a cumulative score of 32 or more points wins the game. Therefore a successful bid of Gros Charlemagne wins the game whatever your previous score.
In the St. John River valley of northern Maine, USA, a different version of Charlemagne is played, without Jokers and with no kitty. The deck has just 32 cards, from Ace down to 7 in each suit and the players are dealt 8 cards each in batches of 4 at a time.
In this version the Nine of trumps is the top trump, followed by the Big Jack (Jack of trumps), Little Jack (the other Jack of the same colour as trumps), then A, K, Q, 10, 8, 7. The ranking of the non-trump suits is from Ace down to Seven as in the Canadian game described above.
The possible bids are numbers up to 7, Le Petit, High Card No Trump and Charlemagne.
The minimum number bid allowed is 1, but the dealer is compelled to bid 5 if no one else has bid more than 4. Therefore bids below 5 are never played, and are used only to convey information. When bidding a number, the player also names a trump suit along with the bid, for example 5 diamonds, 6 spades and so on. There is no hierarchy of suits: each bid must be numerically higher than the last.
La Petite is the equivalent of Mulot in the Canadian game: the bidder plays alone, partner's hand being discarded face down, and the bidder must lose every trick to succeed. There are no trumps and all suits rank from high to low A-K-Q-J-10-9-8-7.
Next above Le Petit comes High Card No Trump, which is a bid to win all eight tricks playing with a partner but with no trump suit.
The highest bid is Charlemagne, a bid to win eight tricks with a trump suit and a partner. The bidder also has the option to ask partner for a card and play Charlemagne alone. This is done by saying something like "Give me an Ace or a diamond" at your turn to bid. That would signify a bid of Charlemagne with diamonds as trump. The bidder's partner gives a card to the bidder face down and the rest of the partner's hand is discarded face down, along with one unwanted card discarded by the bidder, who must then win all eight tricks playing alone with the named suit as trump. The bidder is only allowed to ask for a trump or Ace, not for other cards, and the partner should then use judgment to give whatever card seems likely to be most useful to the bidder. If the bidder plays alone, the opponents have the option also to exchange a card face down after which one of them plays alone against the lose bidder.
The bidder always leads to the first trick, even in a number bid. Players must follow suit as usual, the Little Jack counting for all purposes as a trump. At any turn to play a player may knock the table as a request to partner to lead the suit of the card played at the next opportunity.
In the absence of a Charlemagne bid, the game is won by the first team with a cumulative score of 21 points.