I run an nginx-powered application and I recently turned my attention to using it over https. This is the module in nginx that does this: http://wiki.nginx.org/HttpSslModule
However, I’m somewhat unclear about what is actually required to run a site over https.
What else is there to do to serve my site over ssl? What is the role of the certificate, and is it a requirement that I purchase it from somewhere?
You need a certificate to prove to your user that the server they’re connected to is indeed the one intended (and not a MITM attacker).
If your server is to be used by a limited number of users to whom you could give a certificate explicitly, you could use a self-signed certificate or create your own certification authority (CA).
Otherwise, if you want your certificate to be recognised by most browsers, you’ll need to get one from a commercial CA.
You should find more details in this answer. You may also be interested in this.