When do I need a fb:app_id or fb:admins?

Including the fb:app_id tag in your HTML HEAD will allow the Facebook scraper to associate the Open Graph entity for that URL with an application. This will allow any admins of that app to view Insights about that URL and any social plugins connected with it.

The fb:admins tag is similar, but allows you to just specify each user ID that you would like to give the permission to do the above.

You can include either of these tags or both, depending on how many people you want to admin the Insights, etc. A single as fb:admins is pretty much a minimum requirement. The rest of the Open Graph tags will still be picked up when people share and like your URL, however it may cause problems in the future, so please include one of the above.

fb:admins is specified like this:
<meta property="fb:admins" content="USER_ID"/>
OR
<meta property="fb:admins" content="USER_ID,USER_ID2,USER_ID3"/>

and fb:app_id like this:
<meta property="fb:app_id" content="APPID"/>

To use the Like Button and have the Open Graph inspect your website, you need an application.

So you need to associate the Like Button with a fb:app_id

If you want other users to see the administration page for your website on Facebook you add fb:admins. So if you are the developer of the application and the website owner there is no need to add fb:admins