Eureka! I got it! The 2 systems are Ubuntu. I went through a couple help sites with this query and someone in the Ubuntu forums clued me in to a program called openssh and other gave me advice on how to use it. Many people were talking about a program called samba, saying it was easy to setup and configure, Well. I guess how easy something is or isn't depends on one's own experience. Samba was not easy for me to work with. But openssh was very simple. I said if I ever got this figured out I would write the How To for it. I don't know that it will be needed by anyone or even adopted by this site but here it is:
How To Set Up a Home Network on Ubuntu 9.04 Through Your Router Using Openssh-Server and Openssh-Client:
This is just the most basic requirements for establishing and using a Home Network Connection, giving all computers equal access to each other. If you need security within your own household this How To does not cover it. If anyone else wants to improve on this How To with correction of any mistakes they find, with more detail or by widening the scope, please feel free to do so. I only ask that you do not lessen the detail in it. I am both ignorant of computers and slightly scared of using them. This is the kind of detailed How To I would like to find when I go searching for a means of doing anything with my systems. I wrote this for the benefit of any other eternal noobs like myself out there.
Chose any one machine to start with and repeat these steps on all machines.
1) Male sure that all Machines are connected to the router and have properly functioning Internet Access.
2) Navigate to System>Administration>Synaptic Package Manager and click to start it.
3) In the Quick Search Box type in openssh. Look through the list below the to find and download BOTH openssh-server and openssh-client.
4) When they have finished installing close Synaptic.
You need just 2 pieces of information now. You need the user name of the account you will be logging into on the other computer and that computers I.P. Address. The user name is the what you log on with when you are sitting at that computer. The I.P. Address of that computer can be found by going to that computer's desktop and using the Network Manager. Network Manager is the double computer icon at the top of the screen, just right of the center.
.........1) Right click on Network Manager
.........2) Open Connection Information and there is your I.P. Address around the middle of the page.
Now to continue...
5) Navigate to Places>Connect to Server... click to open
6) In the box that pops up you must chose the Server Type. In that drop down box chose ssh.
7) The next box you need to worry about is labeled Server: Here you type the user name, the "@" mark and the I.P. Address of the computer you will be logging into. It will look something like this: John@194.286.1.65. That's all you need. The other boxes are options, if you know the information and you want to make use of it fine. If not, don't worry about them. Adding a bookmar is useful but not necessary.
8) Click connect. If the user account on the other computer you are trying to access has a pass word, you will be prompted to enter password.
Now you're in and you have complete access to that other computer. Enjpy.