First off, the coins aren't actually stored in your wallet. Instead, think of it like your wallet contains the keys to unlock a coin and send it. If you make a duplicate key to your car, do you now have two cars? No, of course not.
If you "unlock" some coins on one computer and send them away, the same transaction will show up on the other computer.
Note: the above explanation is extremely simplified. For one, "coins" don't actually exist, just previous transaction records. Also, "unlocking" and spending are not independent operations of each other. The best way to see this in action is to view a transaction on the bitcoin block explorer
Additionally, please DO NOT run the same wallet.dat on two computers at once. The current version of the software won't recognize this, so you could end up corrupting* your wallet. This will be addressed in a future version.
*don't worry, it IS repairable, just not very easily with our current tools; so please don't