Yes there are several methods of doing this. If you just want to transfer files on occasion, use Target Disk Mode:
http://docs.info.apple.com/article.html?artnum=58583
If you want a regular file sharing connection, see this article on building a small network:
http://docs.info.apple.com/article.html?artnum=106658
Even Firewire can be used for networking as well, as this Firewire over IP article explains:
http://docs.info.apple.com/article.html?artnum=152307
Note, if the old Mac Mini is PowerPC (like the Mac Mini G4 in your profile), some things shared between them won't work natively or at all, as this user tip explains:
http://discussions.apple.com/thread.jspa?threadID=435350&tstart=0
Mac Mini's which are PowerPC have 2 USB ports in back, where the Intel models have 4.
If you want to see the contents of the other computer and be logged in as the user on the other computer while being logged in as you on the new Mini, consider using Apple Remote Access, OSXVNC, Chicken of VNC or Timbuktu depending on your power requirements. These programs all give you a window into the other computer.
Finally, if you want the processing power to be distributed between the two machines, there is a Mac OS X server function known as XGrid, which you can ask more about on the Mac OS X server forum, or
Pooch.
As you can see there are various types of slave/master relationships or just plain sharing relationships possible.
Message was edited by: a brody