Bridge can be played by two golfers, one against the other, or two teams of two golfers. We'll use 2-against-2 in our examples, because that is the most common way that Bridge is played.
In Bridge, a set amount of points or money applies to each hole. This amount is agreed upon before the round.
When stepping up to a tee box, one team makes a "bid" on the number of strokes (net or gross - decide beforehand, obviously) they think it will take them to play the hole.
Say you're at a tough par-4. You and your partner bid 11. You are offering a bet (of the set amount) to the other team that your side can play the hole in no more than 11 strokes.
The other side has three options: Take the bet; take the bet and double it; or bid lower than 11. If the other side is confident it can beat 11 strokes, it will bid 10. Then it's back to your team: Take the bet, take the bet and double it, or bid 9 strokes.
If one team takes the bet and doubles it, then the other team has the option of doubling back (meaning that if you're playing for money, carefully consider how much you're playing for because it can add up quickly).
Which team opens the bidding on the first hole is determined randomly. On each ensuing hole, the team that lost the previous hole opens the bidding.