Github is without doubt one of the best things ever happened to open source software development. I, like many other FOSS developers, always use Github to host my projects. And like most of them I thought all of my public projects are by default open source.
What we must understand here is that when we post our work online, it's automatically protected by copyright laws unless we specify otherwise. This rule also applies to all Github repositories, both private and public. Actually, being public or private doesn't make any difference. Github clearly mentions this in their help site.
What happens if I don't choose a license?
You're under no obligation to choose a license. It's your right not to include one with your code or project, but please be aware of the implications. Generally speaking, the absence of a license means that the default copyright laws apply. This means that you retain all rights to your source code and that nobody else may reproduce, distribute, or create derivative works from your work. This might not be what you intend.
I just googled this nice web app http://www.addalicense.com to automatically add license to public repositories. But unfortunately it didn't work when I tried it. Maybe it's a temporary glitch. Anyways, just remember to add a license file next time you create a Github repo.