I have a casascius coin -- how can I sell it? I was given an actual casascius bitcoin minted in 2011 as a tip, and I'd like to know where I can exchange it for US dollars. 

How would I extract the private key from this coin to gain access to the digital bitcoins if I were to sell them online?
You have two routes here:
Sell the physical Bitcoin
Don't remove the sticker on the back. Use a website like localbitcoins to sell it to someone near you.
Some people might want a physical Bitcoin, especially since they're not being made anymore.
Sell the Bitcoin 'contained' within the physical coin.
Peel off the sticker on the back. Inside will be a code that lets you spend the Bitcoin. (I recommend using Blockchain.info as your e-wallet.) You can sell the Bitcoin online or locally.
The remaining brass part is mostly worthless.