I was setting up a new user and I checked the Windows Authentication option and tried to save it, but it still made me specify a username and password. Did I do something wrong?

You still have to enter this information.  When using Windows Authentication, it will literally authenticate off of the Windows OS environment.  The SF user password, though required to enter, will be ignored.   We can look at the BO rules on this later.  However, you still have to provide a user name, which should match the Windows user name.  Once authenticated by Windows, we still have to know which user to go and grab so the permissions can be loaded up.

You can see more about this in the docs:

Security Maintenance -> Users -> User Properties