Mapping:

If you have registered a domain name you can map your Google Site to it. See this help article for instructions.
Note that you can't map to a "naked" domain, such as example.com, you have to map to a sub domain. like www.example.com. To make the naked domain work you have to forward it to the www sub domain.

The condensed version of the help article is:
  1. In your domain configuration create a CNAME record of type "www". that points to ghs.googlehosted.com
  2. In your Google Site configuration add a web address like "www.example.com"

If you have more than one Google Site you can map one of them to a different sub domain. As an example, in my domain, k8oms.net, www.k8oms.net is mapped to my main Google Site. I also have a different Google Site that is mapped to tour.k8oms.net.

Here is how I did it:
  1. In my domain configuration I created a CNAME record of type "tour" that points to ghs.googlehosted.com
  2. In the Google Site configuration of the second site I added the web address "tour.k8oms.net". See the image. Click it for bigger view.

Try it: 

Splash Page:

Some people want their site to have a "Splash Page" that has  no header, sidebar, etc. 
  1. To do this you can create two different sites, one for the splash page and another for the rest of the site that may be configured with header, etc.
  2. Map www.yourdomain.com to the splash page..
  3. Map the main site to a different sub domain, maybe ww2.yourdomain.com 
  4. Put a link to ww2.yourdomain.com on the splash page.
  5. Optionally put a redirect gadget on the splash page.
For an example go to splash.k8oms.net 

Note: I use the 'splash' sub domain only for demo purposes. Normally you would follow the instructions above.

If you do not use a domain please see these instructions by PhilRidout 

Forwarding:

In addition, if your domain registrar allows it you can forward sub-domains anywhere. GoDaddy allows for 252 forwards.
Be sure to forward your naked domain 'yourdomain.com' to 'www.yourdomain.com'.

Try:
bob.k8oms.net  forwards to a site on my ISP
webs.k8oms.net forwards to a site on webs.com
test.k8oms.net forwards to another Google Site
music.k8oms.net forwards to another Google Site mapped to a domain

Note: If you use forwarding only the URL showing on the viewer's browser will change to the target URL. If you use forward with masking the URL will remain "subdomain.yourdomain.com".