How do I create a new address in the Bitcoin-Qt client? I am using a Mac and downloaded Bitcoin-Qt as my wallet, and it has "synchronised". However, I am unable to use this program to generate a new address. I click on "new address", and all it makes me do is enter an address and name it…

Is it safe to use a website like Instawallet to "create" an address for me, and then  use this address to receive coins with Bitcoin-Qt?

If two different people are using the same Bitcoin address, in a client like Bitcoin-QT, and if both receive money to the very same address, how will the network know who "owns" the bit coins associated to that address?

What is this "Private Key"? Is there a private key for every Bitcoin address? Where do I find it?

Lastly, if I want to sell my bitcoins, how does the "Private Key" play into all of this?
The address field is unused and disabled - you can't put anything in even if you wanted. It is probably an artifact of reusing the dialog for adding a new sending address.

So you can't use Instawallet to create an address and import it; and even if such a strange procedure was possible, it would not be safe because then Instawallet would have your private key, which should be kept secret.

Since the address is generated locally using pseudorandom data, it is inconceivably improbable that 2 people will share an address (it is less likely that it will ever happen to anyone than that I will be struck by lightning while writing this answer). In this highly unlikely event, funds in this address can be spent by either of them, whoever gets to first.

Every address has an associated private key, which must be kept secret and is what allows the owner to claim funds in that address. Bitcoin-qt stores private keys in the wallet.dat file in the application data directory.

When sending bitcoins to someone (whether to sell them, donate them or to pay with them), the software internally uses the private key to prove prior ownership of them and announce to the network that they shall now belong to the owner of address X. You don't give your own private key to anyone; the new owner will use his own key to send them from his address.