I am trying to set up my paypal account for express checkout. It's asking me for the Third Party Permission username. What do I type in for that?
It sounds like you're in the wrong place in the Paypal settings. There is no "third party permission username".
Take a look at the instructions to set up Paypal Express with Store:
PayPal Express Checkout requires an API Username, Password, and Signature. These are different from your PayPal account details. You can obtain your API details by logging in to your PayPal account, and clicking Profile > My Selling Tools > API Access > Request/View API Credentials > Request API Signature.