All of the following directions are assuming the initial step of your computer being connected to the Internet. If your machine is not connected to the Internet or you do not have Internet access, then connecting to the remote host will fail.
To connect to the remote hosts like TerpConnect, on or off campus you can use ssh or sftp. SSH & SFTP sends all information like your username and password encrypted over the network. This cuts down on the chances that your information could be snooped.
Mac OS X comes with a built-in Using SSH with Terminal in Mac OS X that supports SSH. To connect, open the terminal application. Then, type:
ssh site-name.