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?

I expect it looks in the settings accessed by control panel/network connections

Your OS has knowledge of the interfaces on the local computer and of
the destinations reachable by each of them.

http://www.rfc-editor.org/rfc/rfc1123.txt
http://www.rfc-editor.org/rfc/rfc1123.txt

http://www.kohala.com/start/tcpipiv1.html

Exactly how user s/w obtains this from the OS is likely to depend on
the implementation.

Oops - one of those links should have been
http://www.rfc-editor.org/rfc/rfc1122.txt

If you are on a MS Windows platform (around 2000 and later) then the
answer lies with the computer's 'Registry', a database in which all
programs can register their own existance (and data), in what are
called keys.

To kick up the Registry in a window so that you can see it you have to
start at the Command Prompt (usually found by pressing
'Start/Programs/Accesories/Command Prompt'. Once you have the Command
Prompt utility in front of you, just type in 'Regedit' and press
enter.

When the Registry pops up in front of you, you will see a series of
'Keys' that programmers know how to navigate programatically. Click
around it to your heart's contemt, but don't delete any keys, or
change any values. You can try the 'search' feature by putting in the
word 'Internet' and see what keys it shows you! (Don't worry about the
search feature being very slow.)

An advanced user would be able to, for example, find the keys that
relate to your other two internet programs, who by deleting the keys,
would prevent other programs from displaying their annoying messages.
Better off, uninstall the offending programs by pressing
'Start/Settings/Control Panel' then select the 'Add/Remove Software'
option.