Bitcoin is a fascinating subject both from the technical and social viewpoints. While there is a lot of people online talking about how to use it and whether you should use it, there seems to be far fewer people talking about how the wallet works from a backup perspective. This seems strange as your wallet contains your money and if you do not understand how it works you might loose your money.
If you don't care about the WHY and just want the answer jump to the bottom of this post.
The first thing to understand is that your wallet does not contain the actual money. Your wallet contains a lot of public/private key pairs (I'm assuming you understand the basics of public/private key cryptography - if not I'll be writing something up on that soon inspired by the latest portforwardpodcast). The blockchain contains the list of who has which bitcoins.
Every time you are sent money it is sent to one of your public keys and only your private key is able to send that money on again. As your bitcoin client downloads each new block it compares it with the keys in your wallet and adds or subtracts the amounts sent to/from your keys.
This is part of the security of Bitcoin as if someone tries to send you bitcoins your client can see if they actually have those bitcoins to send by looking back through the blockchain. If everything is correct then the transaction will be included in the next block generated which is the first confirmation. By the time you have 6 confirmations you can be pretty sure the money is yours.
The public/private keys can become used in two ways so your wallet.dat file contains 100 spare keys for future use (you can change this default to anything you want). A new public/private key pair is automatically generated each time you send or receive bitcoins to help improve the anonymity of bitcoin. Each new key pair is added to the end of the list and every time a new key pair is needed the oldest spare one is used.
You can see this happening when you receive bitcoins - the client shows your current bitcoin address on its main window. When you receive some bitcoins a new address from the list of spares will become active and a new address will be generated and added to the end of the spare list. Since your wallet still contains the old address you can still receive funds sent to it without any problems.
When you send bitcoins a key pair can also be used. This is because if you have received 10 bitcoins to a single address and you want to send 2 bitcoins the transaction sends all 10. 2 bitcoins go to the person you are paying and the remaining 8 bitcoins are sent back to a new one of your own addresses. This is all hidden by the client so you do not see these keys.
You can see this happening at http://bitcoincharts.com/bitcoin/ which shows the latest unconfirmed transactions - almost every transaction has 2 outputs. One will be the money going to its new owner and the other will be the remainder of the money going back to its existing owner on a new address.
You can backup your wallet.dat file, complete several transactions and keep using the new addresses to receive money with no problems. At some point, probably about 100 transactions (sending and receiving) you will start using key pairs that are not in your backed up wallet.dat file. If you then loose your wallet you could loose a lot of your bitcoins as the backup will not have valid key pairs to access the money.
What this means for your backup of wallet.dat:
So taking all this into account you should backup your wallet.dat file regularly but it is not necessary to backup after every single transaction. Your wallet already knows the encryption keys it will use for the next 100 transactions. Speaking of encryption please encrypt your wallet! A future bitcoin client is expected to include encryption which should improve things but until then its your money - look after it!
On Linux your wallet.dat file is stored in ~/.bitcoin/
On Windows the location of your wallet depends on your windows version. For Windows7 (and Vista if you are still suffering that OS) it is in C:\Users\<your user name>\Appdata\Roaming\BitCoin. For older versions of Windows (eg XP, 2003 etc) it is in C:\Documents and Settings\<your user name>\Application data\BitCoin.
Just close the bitcoin client and copy this file somewhere safe (both "safe if my computer dies" and "Safe from anyone that wants to steal my money"!) The only file you need to backup is the wallet.dat file - all the other files will be re-created/downloaded if you start the bitcoin client with the wallet.dat file in the bitcoin data directory.
Disclaimer: Everything here is based on my understanding of how things work - I am in no way connected with bitcoin apart from being a user. The main bitcoin client is still at version 0.3.24 - it is a relatively new program that is likely to change as new issues with the growing bitcoin network emerge and new features are added.