if I run a .swf on the server with a remote object source = "server_path.mycfc"

what would source be if I was connecting to an external server at say
www.mysite.com? Would it be

mysite.com.server_path.mycfc

It is going to be the path to the cfc starting at the root of the site to the
CFC unless there is a ColdFusion server mapping that overrides it. However I
have not tried this using mappings and I hear from others it does not work well.

So lets say you had a cfc that resided in this folder starting from the root
of your site

{siteRoot}/com/mydomain/cfcs/

The source would be com.mydomain.cfcs

And the destination would be ColdFusion (unless you had a need to configure a
different destination in the XML files on the server.)

So to answer your question more exactly a fully qualified domain name such as
www.mysite.com simply points to the root of your site. The Source then states
starting from the root where is the CFC I am wanting to talk to.

Where it gets confusing is the habit of CF developers to place all their cfc's
into folder paths using the Reverse domain name like the above
{siteRoot}/com/mydomain/cfcs/.

If you were to see this exact path in a server log however the way it might
look would be

http://www.mydomain.com/com/mydomain/cfcs/mycfc.cfc

Of course you will never see that in a log because of the nature of cfc's but
the point is to show how the path / source works.

Hope that helps