Q:

How does the wallet work, online and offline? 

I've been reading about bitcoins and I've come across a few questions:

How does the wallet actually work?  From what I can gather, I use the wallet as a centralized service to store recieved coins, and to send coins.  If recieved coins are stored physically in the .dat file, what are the safeguards that prevent users from duplicating the .dat file and getting more coins? 
Do I need to have the wallet client on to receive transactions?  In other words, if i don't leave the wallet client active on my computer, will i be able to receive transactions the next time i open the wallet? 

Cheers. 

A:

Although wallets indeed contain your "coins", the ability to spend them is tracked by the network.  Every fully validating node keeps a list of available coins on the network.  They do not know who has which, only which (Bitcoin, not IP) address (or addresses) they are associated to.  In case you try to spend the same coin twice - for example because you copied your wallet to another computer, and then tried to send a transaction from each - you may temporarily confuse the network.  Some nodes may first see the one transactions and some others will see the others first.  However, the strength of the system is that this disagreement cannot exist for long and only one of them will be accepted after some time. 
The essential thing in your wallet is not the transactions, but the keys.  Each address has an associated public and private key.  These private keys never leave your wallet file and are necessary to spend the coins assigned to your addresses. 
Because the transactions are tracked by the network, you indeed don't need to be online to receive transactions.  The next time you connect, your client will ask the network for up-to-date information.  During the synchronization that follows, your incoming transaction will be fetched, and the client will detect this as an incoming payment.