Suppose I have an online Bitcoin wallet. Now, someone breaks into my computer and steals my Bitcoin wallet.

It is generally not recommended to store Bitcoin in any online wallet or exchange. Mostly because it is unlikely that you are in full control of your private keys. In the event the website is compromised, hacked, or you become victim of some sort of man-in-the-middle attack, you are at risk of losing your funds. This is especially true if the service is poorly managed (Mt. Gox had a similar fate).

Fortunately I have a backup of my wallet on another computer. Now, What determines who has the bitcoins now?

Anyone who obtains your private key(s) have full control of your funds. The only variable in this case is time (who can spend the funds faster). If you are even slightly concerned your keys may have been stolen but have not yet been spent, you should defiantly move them out of that wallet and into a freshly generated one.