Quick Method Using Godaddy
>Go to your Godaddy Account and click on My Account
>Find the domain you want to use and click LAUNCH
>Under Domain Forwarding open the dialog and Forward your domain to the subdomain www.yourdomainexample.com and click save.
>Now launch the DNS Panel and scroll down to where you see CNAME entries
>add a new CNAME entry and add a new one for www and point it to ghs.google.com and click save.
Now finally come back to your sites backend by clicking More>Manage Site>Web Address and enter www.yourdomainexample.com as a new address and save..
Other Methods (Different than Godaddy)
These instructions tell you how to make your Google Site appear at your own custom web address, e.g. how to make your Google Site appear at www.example.com. Your Google Site will always appear at the sites.google.com address as you need to edit the site there and if you set your site's visibility to public viewers can access it at the sites.google.com address also. If you map a web address to a private site Google Sites always redirects you to the sites.google.com address as it needs a secure connection to authenticate the users.
Google Sites already provide instructions on this at http://support.google.com/sites/bin/answer.py?hl=en&answer=99448, but many are unable to follow these instructions for whatever reason. Also bcj19 has written an equally detailed step-by-step domain mapping guide on his Unofficial Google Sites Help site, which is well worth a read: it will give another perspective on the process.
You need to edit your DNS records to make this work so if your domain provider does not allow you to edit your DNS records, or add CNAME records, then you cannot follow these instructions. If your domain provider allows you to change your nameservers then you can sign up for a free ZoneEdit account to get editable DNS records.
You need to do different things if you're using a Personal Google Site (a site that's at http://sites.google.com/site/sitename) and a Google Site at a Google Apps domain (a site that's at http://sites.google.com/a/example.com/sitename), so I have split the answer in to two: