Summary: Once Windows Networking is enabled on your local network it's relatively easy to share printers as well as files. We'll walk through the steps.
I have a wireless network with 2 computers ( A & B ) and a printer ( A1 & B1 ) physically connected to each computer. A is connected to A1 and B is connected to B1. My question is can I use computer A to print to printer B1 or computer B to print to printer A1? I do not want to make the printers wireless at all.
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The short answer is yes.
Assuming that computer "A" and computer "B" can "see" each other across your network, it's actually a relatively simply matter to print on either printer from either machine.
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You may note that one of the things I don't spend a lot of time discussing here on Ask Leo! is Windows Networking. Getting two Windows machines to see each other and successfully share files can at times be incredibly frustrating. When it just works (which it often does) it's great. But if you encounter problems there are so many things that could be wrong, it's just not a topic I take on very often.
So, I'm going to start by assuming that your machines A and B can "see" each other. By that I mean that in Windows Explorer, in your "Network Neighborhood" or "Microsoft Windows Network" you see the other machine listed, along with any file shares it might have:

The trick is simply that you can share a printer on your network just like you can share a folder.
On the machine with a printer physically connected to it (machine "A" with printer "A1", to use your terms):
open Control Panel
double click on Printers and Faxes
right-click on the printer that's physically connected to the machine
Click on the Sharing... item. This will open the printer's properties dialog opened on the Sharing tab:

Make sure that Share this printer is selected, and give your printer a name. As you can see I've given my printer (what you've called "A1") a name of "hppsc131".
Click OK.
In order to use this printer from another machine you'll need to add it as an available print.
On that other machine, your machine "B":
open Control Panel
Click on Add a printer to start the Add Printer Wizard
Click on Next and you should see a dialog similar to this:

Make sure the "network printer" option is selected, and click Next. You'll get a dialog similar to this:

You can, if you like, try browsing for a printer. Windows will "look around" to try and locate any printers that are shared by any machines on the local network.
I find it more reliable to enter the printer's name directly. In my example above, the computer name is "Leo" (what you've called "A"), and the printer name I chose was "hppsc131" (what you've called "A1"), so I would enter "\\leo\hppsc131" as the Name: after selecting "Connect to this printer":

Click Next
The next page of the wizard will ask if you want to use this newly connected printer as the default printer. Select Yes or No as you see fit and click Next.
The final page will summarize what you've just done.
That's really all there is to it. Now, on this machine B I can print to the printer on machine A.
Since you've indicated that you also have a printer on machine B that you want to print to from machine A, you'd simply repeat this process again, this time sharing the printer on B and connecting to it from A.
Related:
Ask Leo! - Why does my Microsoft Word document display differently on different computers?
Ask Leo! - Why does my computer 'Find New Hardware' for my printer every time I boot?
Ask Leo! - Networking Sucks - podcast.
Article C3047 - June 5, 2007
When trying to add a printer to my network so the remote laptop can print from it, I can't expand the network and see the printer that is wired to desktop?
Posted by: Mark Hull at March 12, 2008 8:29 PMHow do you do this on two macs? [one macbook and one imac, if it matters]
Posted by: David at March 18, 2009 8:10 PMBoth my computers on the network were named by the programs. Both are XP. I can print from my newer computer to a DellAiO in the bedroom on an older computer, but I cannot print from my new Kodak printer on my newer computer to the Dell in the bedroom. It browses and finds the printer and when I click next, a box comes up and says the name is either wrong or the printer is not connected to the server which it is if the browsing found it.
Posted by: Wanda Grahe at May 17, 2009 8:10 PMIs it possible to use printer network with two different computers ( 1 Macbook and 1 Asus notebook)? Both units are wifi ready.
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Posted by: samsung 315 at August 9, 2009 5:44 AMFor consumer friendly articles visit http://www.samsung315.co.uk
Why does my wife's laptop no longer connect to the wireless printer?
Posted by: Robin Frost at August 9, 2009 6:47 PMWe have a wirless router. I have a laptop (running Windows XP) hard-wired via a docking station to 2 printers. I have enabled sharing on both printers. My wife's laptop (runing Windows Vista) has been printing to these 2 fine. Tonight, it now says we need to install the print drivers (for both printers). I tried physically connecting one of the printers and it installed fine, saying "You already have the driver for this printer installed" and could print no problem. So, I suspect it's a network issue not a printer issue? What could be the problem? I can "see" both printers in windows explorer on my wife's laptop, I have checked that sharing is enabled, it always ends up asking for the driver and then giving an Windows cannot connect to the printer" error message. Any other pointers? (Sorry it's kind of open ended, let me know what more I should add to make the question easier to answer?)
Thanks for all your help!
I did as described and I get "The server for the printer does not have the correct driver installed." I search for the correct driver, but cannot find it.
Please help
Posted by: Scot at August 30, 2009 1:54 PMScot