A GitHub account is not required for anything. It's just a convenience and offers a little extra functionality. If you connect your GitHub account to your CodePen account, you can use it to log in, and to export Gists of your Pens to your GitHub account.
You can connect and disconnect GitHub accounts at anytime.
Under your Settings > Account (while logged in) you'll see a section for GitHub:
If you're currently connected, the button will offer to unconnect it for you. If you're current unconnected, you can use it to connect. Connecting takes you to GitHub.com for authentication. You can connect to any GitHub account you wish (one at a time).
Log In with GitHub
Our Log In screen offers a "Log In with GitHub" button. You may use that if your CodePen account has a connected GitHub account. If it doesn't, we'll let you know when you try to use it.
Other Features of GitHub Auth
There is only one right now.
When you export a Pen as a GitHub Gist the Gist will be under your GitHub account, rather than anonymous. This makes it all the easier to find again, and even better, if you export the same Pen as a Gist again, it will update that Gist rather than make a new one. That gives you the ability to see difs (what has changed) through the Gist interface.