I understood the bidding a little.
The basics involve bridge being played with two teams of two players each, sitting opposite each other. Each player starts with 13 cards (from all 52) dealt out by the dealer starting with the player to his left (I think), and each player can see their own cards only.
From what I gather the bidding involves one member of the partnership announcing how many tricks they can win with one suit depending on which cards they have, then waiting to see their partner's response, depending on the strength of their own hand, and developing their own bid further, the other team of course responding with bids based on the strength of their hand, or, if they have no such luck with their hand, or declare "no bid" and play no further part in the bidding process. In other words, the bid one makes sends a message to their partner about how good their hand is.
Say, if they have a lot of cards of the Hearts suit, no matter what rank, they might open with "2 hearts" meaning they could easily win two tricks if Hearts were the trump suit. The number might be higher if they had several high-ranking hearts, of course.
If they had a lot of high-ranking cards with no one suit dominating, then they might bid say "3 no trump" meaning that they could win 3 tricks with no one suit acting as a trump suit i.e. the tricks would be won by the highest ranking card of a given suit being played only.
During the bidding, I think the suits are ranked clubs < diamonds < hearts < spades < no trump, and go up by 1 until all other players agree to "no bid", then the game starts with the player making the final bid trying to fulfil his contract of number of tricks (i.e. what his final bid entailed, be it "7 hearts" or whatever). (His partner then displays his whole hand to the table during the game, acting as a "dummy" for the main player for that game.) The other two players then have to use their skill to try and ensure that the main player's contract is not kept by winning tricks themselves.
Bluffing during the bidding process is considered cheating, from what I recall. Plus it doesn't really help your bridge partner.
The winning of tricks is pretty basic stuff - highest ranked card of a suit displayed by the opening player wins, unless trumps are involved in the contract for that particular game, in which case the highest ranked trump card wins the trick. And again, there can be no holding-back cards of a certain suit if played by the opening player during that round - if you have a spade if a spade is played at first, you must play it. As a result, trump suit cards are usually not played until the third or fourth trick, although in a no-trumps game crazy things can happen, with tricks won for low ranking cards simply because the other players can't match that first card's suit. (One funny game I remember involved one man trying to cheat by giving himself all the cards of one suit, and declaring "7 hearts" as an opening bid, to which his opponent replied "7 no trump". The cheater lost, badly.

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The scoring I still don't know much about.