I want to know how a computer or, more specifically, a piece of
software, knows if there is a connection to the Internet and how it
finds it.  When some software gets installed on my PC, it sometimes
says something like:

 ?2 connections to the Internet discovered.  Bellsouth dialup
connection and fast connection through local area network.  Which do
you want to use??

Where does the program go to find out what Internet connections exist
on the computer?  Does it send a ping or something out of the port to
make sure that the connection works?

Which Operating System you are talking about?
If windows, may be this information is obtained from Windows Registry.