How to play bridge
Bridge is a partnership card game for four people commonly referred to as North and South playing against East and West. When you're playing bridge online against the computer you’ll normally sit South and the computer will fill in for the other three.

The Auction - choosing a contract
There is an auction at the start of every hand. During the auction, players make bids to tell their partner what suit they like best. For now, pick a suit you like and bid that - you'll do fine!
The terminology for bidding is always a number plus a suit. For example, 1♥ or 4♠.

Those hearts look pretty good so South bids 1♥.
Each bid must either be for a greater number of tricks or a higher ranking trump suit, so we need to know the rank of the suits.

After your 1♥ bid, if West wants to bid spades, he can bid 1♠ because spades rank higher than hearts but if he wanted to bid clubs he'd have to bid 2♣.

West has made spades the trump suit.
You can either bid...

North likes hearts, too!
...or pass.

East has nothing to say.
Going, going , gone!
After three passes in a row the auction is over.

Going, going, gone. Auction over.
2♥ was the last bid so hearts is the trump suit and 2♥ is the contract. That's where the name contract bridge comes from.
Playing the cards
South bid the hearts first on this hand so South is the "declarer".
The opening lead
After the auction is finished, the person to the left of declarer makes the opening lead.

West leads the ♠6.
Declarer and Dummy
Declarer's partner is called the dummy. Dummy's cards are placed face up on the table and declarer takes control of that hand.

North is the dummy on this hand.
Following suit
In clockwise rotation, everyone plays a card and if possible must 'follow suit', which means playing a card of the same suit as the one led. If you can't follow suit any card is ok.

North follows suit with the ♠3.
Winning Tricks
Each round of four cards is called a trick. The highest card of the suit led wins the trick unless someone plays a trump in which case the highest trump wins the trick.

South wins the trick with the ♠A.
The person who won the trick plays the first card to the next trick.
Once you've finished you can calculate the score.