The mingetty processes are the processes that allow you to logon to the text consoles in Linux.
There are 3 or more because you've got this number of consoles configured - you can switch between them with CTRL-ALT-Fn where n is the number of theconsole you want. Your X Console is normally on the one after the last text console. If you're not on a X Console ALT-Fn works as well.
They are started by the init process - look at /etc/inittab to see the lines launching them.
The name is short for "minimal get teletypewriter", which shows the age of the technology...
To see connections to your machine, use netstat. Try "netstat -tn" in a console window. You will see a list of IP connections by number currently extant. Use netstat -t if you want the names translated to text form (this is slower). Not all connection types are shown in this list, however - e.g. masqueraded connections.
For both commands, read the manual pages for more details - use "man (command)" or "xman".