How does Stack Overflow allow new users to login into Stack Overflow using their Gmail or Facebook or Yahoo accounts? I am working on a little project website and I was curious how Stack Overflow does this. Do I need to get permission from these companies to do this or can I just do it?
StackOverflow (and all other StackExchange sites) use OpenID.
OpenID is a decentralized authentication protocol that makes it easy for people to sign up and access web accounts.
Yahoo, Google and facebook are all OpenID providers, so simply implementing it on your site will be enough for your users to be able to login using them (and any other OpenID provider).
There is no need to get permission - the permission will be gotten by the users of your application.