What are the sites where I can make my own website for free and have my domain without pay?

See InMotion for a free domain forever...as long as you buy a hosting package from them. Otherwise, you'll have to settle for free sub-domains.

Free web site sources:

Budget Web Hosting List (2 pages): http://www.thefreecountry.com/webhosting/budget1.shtml
http://www.piczo.com
http://sites.google.com
http://homepagenow.com/
http://www.angelfire.lycos.com/
http://WebStarts.com
http://freehostia.com/
http://bravenet.com/
http://www.sitesled.com/
http://www.blinkz.com/

There's no such website.

You will not get a name.com for free. PERIOD. You can however get a Your-bit.name.com for free.
For that I would try 000webhost.
IF you do need a name.com just go to godaddy use a promo-code like GB1/TZD2 to get a discount and then change the NAME servers to the ones at 000webhost.

The best website maker is Webs (formerly Freewebs)
Easy to build, calander, security, Email List, Great Support, Photo Albums, Videos, Blogs, forums, games, apps, widgets, customizable HTML, 50mb space, LOADS of templates and a members login and you can send messages to them and they can send messages between themselfes, and add them as friends. You can also send emails to all your members at one and you can do templates and a very easy site builder. Theres no need to learn HTML!
Honestly, I have built 4 websites, and they are absolutely fabulous!
You can also use your own templates

have a look at my non-profit: http://www.eastrosevillescouts.com (i paid for the domain, but if you dont want to, you can have a www.[yoursite].webs.com

Have a great time, I know you will LOVE it

The only way you might be able to get a free domain is if you're willing to pay for hosting services. Some web hosts will throw in a free domain if you signup for hosting services (some starting at US$4.95/month).

If you want your website for free then you'll have to pay for your domain name. However, one advice, if you're using the website for your business, you might want to stay away from free website/hosting services. Many free hosts will plaster their ad on your free website - it'll make your business website look unprofessional.

If you want affordable domain + website service, check out Doteasy.com - you can get a domain name and basic web hosting services (with free website creator and email accounts, etc.) for US$25/year. It's banner-free hosting, too.

Eh. I have never seen that before getting a free domain. If you are making it for a business you should invest a little money into it.