Edit - took me so long typing this MrSaffron got in first, but I think the two posts add to each other so leaving it here. End of edit.
Yes. That's what routers are for.
Everybody tends to be a bit slipshod, just like we are in speech, when we all know what we are talking about. But not everyone does know at first

.
What we call a router is these days usually an ADSL router. That means a combination box, like your O2 Box, containing a router, an ADSL modem, and an ethernet switch - usually 4-port like yours. It can be wireless capable or not, the difference being whether it has a fifth component - a Wireless Access Point. Again, yours has.
So the modem connects to the internet and computers connect either to an ethernet port on the switch or to the Wireless Access Point.
The modem, switch and WAP connect to the router.
(There is also a thing called a hub. That is an earlier piece of kit that is similar in function to a switch, but less efficient. Some ADSL routers apparently contain those instead. The issue is complicated by the fact that I believe the BT Home Hub does not contain a "technical" hub, it contains a switch. It is using the "english" meaning of the word "hub", not the computer industry technical one).
The router looks at the IP address it gets a message for and sends it out on the appropriate connection, to the modem, switch or WAP. The switch does a similar trick, transferring messages between computers connected to one of its ports, or to the router.
So your setup is fine. Your next step is to enable sharing of one or more folders on each computer. Look it up in Windows
Help. By default Windows stops other computers on your network (LAN - Local Area Network), accessing everything on each other, for fairly obvious reasons), so that you have control over what you make visible.
When setting up sharing I advise disconnecting from the internet, so that you can safely disable your firewall software. This can make things far easier at first, as when you turn the firewall on again you might find the networking fails. At least you then know that's where the problem is, whereas otherwise you can literally spend hours wondering why it doesn't work when you have followed all the Windows Help instructions.
You would have to go into the settings for the firewall to allow traffic between the computers you have, but ask more about that if it happens.
Once you have networking working without the firewall then even if it fails when you turn the firewall on it is safe to reconnect to the internet, (
with the firewall on). The computers will both continue to talk to the world, just maybe not to each other. But you can come back for help on that and configure the firewall as necessary.
My broadband basic info/help site - www.robertos.me.uk
My domains,website and mail hosting - Tsohost.
Internet connection - O2 Standard.
Edited by RobertoS (Fri 29-Oct-10 09:59:58)