These instructions tell you how to insert (embed) a Google Calendar into a Google Site.  Many people make the mistake of trying to use the embed code from Google Calendar and try to insert it into a Google Site - this is incorrect - please follow the instructions below.

Note:  If you have a public site you may wish to create a separate public Google Calendar first, so none of your private appointments appear.  Remember to make this Google Calendar public.

Instructions

1. Go to your Google Calendar and go to its settings - see screen shot
   Google Calendar Settings Menu.
2. Use the blue HTML icon in the Calendar Address section to make the Calendar Address appear - see screen shot
   Google Calendar HTML Icon.
3. Copy the Calendar Address not the Private Address - see screen shot
   Google Calendar Address.
   
4. Go to your Google Site and edit the page you wish to add the Google Calendar to: once in the editor use the Insert button and choose the Calendar menu entry - see screen shot
   Google Sites Insert Calendar.
5. Paste the Calendar Address into the Calendar Address box, set any settings you want and use the Save button to insert the Google Calendar - see screen shot
   Google Sites Calendar Settings.
   
6. Once back in the editor use the Save button to save the changes and view the inserted Google Calendar.

Now events added to the Google Calendar will appear in the embedded calendar in your Google Site.

Video

I also include a video of how to insert a Google Calendar into a Google Site, including how to make a new public calendar.