I understand that securing your bit coins is basically about protecting the private keys for your addresses. This makes sense from my past dealings with public/private key encryption. How do exchanges handle and store your bit coins for you? Is it basically a secured list of your keys? Can you get access to the keys either to back them up, spend them outside the exchange, or move them to another exchange? What would happen if an exchange was compromised (i.e. siezed). I can imagine that I'd lose any dollars but my bitcoins are still accessible to me outside the exchange as long as I have the private keys, right?
You access the bitcoins in your account at an exchange through their web interface only. They keep the private keys to those those bitcoin addresses hosted at their exchange. So, even though its your account, they control the private keys. If you want to move those funds outside the exchange, pay them to yourself on a wallet hosted on a device you control. An exchange is unlikely to give you access to the private keys, because then the private key would be out of their control and they could not guarantee that the funds were safe nor be held responsible if they were spent.