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How old are those Macs ? Are they connected to same network ?
They are both 6 years old. On the same network.
Note: I am having problems with both computers and may occasionally be interrupted briefly.
There are mainly two ways to transfer the files between. One is through Firewire Target Disk Mode and another is by Apple File Sharing. If they are connected on same network, then you can enable Sharing on both Macs and transfer the files using your network.
See here --> https://support.apple.com/en-ap/HT204445
(The instructions are for latest OS X, but it is the same procedure for 10.6 Snow Leopard)
or see here --> http://www.dummies.com/how-to/content/how-to-enable-file-sharing-on-mac-os-x-snow-leopar.html
Another way is, get a Firewire cable and you can connect both Macs with Firewire cable and then use Target Disk Mode to transfer. See here- > https://support.apple.com/en-in/HT201462
Target Disk Mode will allow you to mount the hard drive of one Mac to another Mac and you can copy files from it just like an external hard drive.
If you have any questions or would like any additional information, please don't hesitate to ask.