Most people do not offer free top level domain names (i.e. .com, .net)...in fact, you will be quite lucky if you find one. Free subdomains (i.e. yourname.website.com) are available readily, as are free domain names such as co.cc, etc. There are many of those out there.
Many web hosting companies offer free domain name with one or more of their web hosting packages. However, you need to purchase the hosting plan in order to get the free domain. Also, many of the web hosting companies will have conditions or limitations that you must read.
There are many web hosting providers offering Free domain name within their hosting plans, but you should pay for the hosting for the whole year in order to get free domain.
It's 2$ for a .info domain on GoDaddy (first year only, of course). Really, a coffee costs more than that, so you should just get it. You'll benefit from learning what domain management is all about. Or you could get a free subdomain with Google's AppEngine (something like "your_website_name.appengine.com). That might be good enough.
They do domain forwarding. You don't own it, they just redirect to your site if someone enters that url. If someone registers that domain, it will no longer work.
Free domain is not suitable for e-business. you can use these types of domain only without any guarantee. The best choice is choose a better web hosting company and by a hosting plan they offer you to buy a free domain name. Refer some web hosting review sites.
You can try blogspot, weebly and wordpress. then you can redirect it to a co.cc domain. For example, if you have a wordpress site (yourdomain.wordpress.com) you can use co.cc to make it (yourdomain.co.cc) Cheers!
That is good idea so far. But every moment you may lose all what you have and gained as you are dealing with free service. So sometimes that worth to pay few $$ and be sure that everything will be fine.
Try to look around for some kinda promotional offers, i have recently stumbled upon a forum which was running a contest and they were saying to award a free domain to the winner. if you just search around, you might find some promotional offers which might help you to get a free domain name.