Note: these instructions are for Windows XP. I don't know if Vista handles sharing differently, but it's likely you'll need to be logged in with admin rights as a minimum.
Step 1: Connect both computers to router.
Step 2: Go to computer with stuff you want to copy.
Step 3: Open Network Connections (may be under Settings), find your LAN connection (usually says "Local Area Connection"), right-click, Properties.
Step 4: Under "This connection uses the following items:", make sure that "File and Printer Sharing for Microsoft Networks" is listed and checked. If not, add it under Install->Services->Microsoft.
Step 5: Open My Computer, find the folder you want to share.
Step 6: Right-click on the folder, click Sharing and Security...
Step 7: Check "Share this folder on the network" and give it an easy to use name. Do not check "Allow network users to change my files".
Step 8: Make sure your firewall is configured to allow file and printer sharing - if you're using Windows Firewall, there will be an indication of the status on that dialog. If you're using a different firewall, check its settings.
Step 9: Click OK. If you haven't set up sharing before, Windows may need to think for a while.
Step 10: Go to the computer you want to copy files to.
Step 11: Make sure file and printer sharing is enabled, as above.
Step 12: Open My Computer, go to My Network Places (may be under Desktop), Entire Network, Microsoft Windows Network, then the workgroup or domain your first computer is in, then the computer name itself.
Step 13: If things are set up properly, the folder you shared will be visible under that computer's name. Copy the folder itself or files from that folder like you would any other file/folder.
Step 14: If you can't see the share, try restarting both computers. If you still can't see it, you may need to configure the network names to be on the same workgroup. This is done from My Computer->Properties->Computer Name. Both machines should have the same workgroup name and should also have distinct computer names.
Step 15: If you still can't see the other computer on the network, try typing its share name manually. For example, if it's named "gnomer01", open My Computer, go to the Address bar, and type "\\gnomer01". You should see the machine. If you don't, you'll need further troubleshooting.