I installed Pegasus e-mail, which was quite simple, but I get an error message when I try to download the messages, which states that "A network error occurred during connection to the host".
I have edited the POP3 tab several times to no avail. Any suggestions ?

An Internet Session Log should help identify the point of failure. To created a log, go to Tools > Internet Options then tick the "Create Internet session logs..." tickbox at the bottom of the General Tab. Now do a check for new mail (use the "Check you POP3 host for new mail" button in the toolbar). The resulting log file will be created in the mailbox directory, in a subdirectory named TCPLogs. The filename extension identifies the internet session that was logged but these are plain text files so can be opened with a text editor. IMPORTANT: DO NOT POST THE ENTIRE CONTENT OF A LOG FILE. These log files contain your authentication credentials, sometimes in plain text, sometimes encoded, but that coding is easily decodable. If you feel the need to post the log file, first remove any entry that starts with "USER" or "PASS", or the four entries following an "AUTH LOGIN" entry.

You might also post the configuration settings you are using. Someone here may be using the same service and could spot the problem.

Server host name:
Is username in form of Username only or is it your full email address.
Don't post password
Server TCP/IP port:
In the Security tab, which option is ticked?