Remember that online wallets are more vulnerable, because you have to trust a third part, and the sites could be hacked. With bitcoin-qt, you store your BTC on your computer and not on the internet. But with this you have to remember a bcakup of your wallet on a usb stick or smth, in case your machine crashes.
No , you don't store BTC on your computer.
You store the keys that make possible to spend those BTC.
If anybody gets your private keys they can spend them .