To be able to put relevant AdSense ads on your Google Sites, your site must be Public.
If your site is Private and you try to monetize it, you will see a warning similar to this:
WARNING: This is a private site and AdSense requires that the site be publicly accessible in order to serve targeted ads. If you enable AdSense on a privately accessible site, the ads that are shown will not be based on the content of your site and may only be Public Service Announcements.
However you can still go ahead and monetize (add AdSense ads) to your site as long as you don't mind the Public Service Announcements for now and that you remember later to make your site Public (otherwise no one will see you!).
(See how to make your Google Site Public or Private or with specific access settings)
And you must have a (free) AdSense Account.
(Read about AdSense in common sense English and how to set up an AdSense Account)
More Actions > Manage Site > Site Settings > Monetize > Monetize this Site (button)
Only after you have done that, will the second part of the page become activated.
Under Display Ads > Site Sidebar, the "Add" button is now activated. Adding ads to your site sidebar will make it global - i.e. Google Adsense ads will appear on all your pages. Google will take care of displaying relevant ads for each page. All the work is done for you :)
Additionally you can add more AdSense ads into specific pages. You choose where and on what pages.
While on any page you choose, and where specifically on the page > Insert > AdSense
Check on maximum AdSense ads allowed per page
Now all that is well and good:
There is one more (optional) piece: you may want to track how much earnings are coming from this google site.
This just means you want Adsense to be able to tell you how much earnings is coming from this one site. For Adsense to be able to do that, you need to give the URL for this site to Adsense and tell it to track it for performance.
To set up Adsense so that it can track your Adsense earnings from your google site: