For the full API documentation go to https://bitpay.com/downloads/bitpayApi.pdf
In a nutshell:
1. You make an API call to bitpay telling it how much you want to charge, what currency to use and your IPN (Instant Payment Notification) url.
2. The API will respond with a URL which you will redirect the customer too.
3. The customer will then make the payment.
4. Bitpay will then call the IPN url you specified in the request letting you know whether the payment was successful or not.
Thanks. So it's almost as easy as integrating a Paypal or Google checkout button into the site, it seems?
Good question. Now perhaps this will sound stupid, but my ultimate vision of this thing is that we will be able to leverage the power of the Bitcoin community and open-source methodology to develop and maintain our site, and I'm planning to hire Bitcoin developers on paid contracts (and pay them in Bitcoin). I made that decision when I realized there were tons of Linux/php devs lurking around here looking for some extra coin. I also felt like I shouldn't let my apprehension about learning new technologies influence my decision.