Free domains and Google Apps

Hi guys,

I am looking for some site in which I could obtain a free domain. Seeking info about this, I have founded GoogleApps, if I haven`t understood bad, you can register your domain free.

But I have the doubt if register domain is the same as obtain it. Explaining better ¿in Google Apps I can obtain the domain, writing its name, or only it is valid to register
when I have obtain/buy the domain?

I know that this question can be a bit messy, but I think I haver the concepts not very clear.

I would appreciate if somebody could help me to solve this question.

Regards and thanks with anticipation