New to OmniDiskSweeper
 

Using a MacBookPro w/ almost-full HD, I ran OmniDiskSweeper and found a 16G "Previous Systems.localized" folder. OmniDS designates this as "Previous Systems.localized 16.0 GB"; no other text indicated.
I would like to delete this file but need to know if it is safe to do so. Can you tell me if I need this folder?

The lower-lever folder is entitled "2008-09-07_1324" looks like a date code?

You must have done a system (re)installation and told it to keep a copy of the previous system. If your new system is working fine and you don't think you'll ever have any need to retrieve something from the old one, you should be able to delete it and get the precious disk space back. Otherwise, you can drill down into that folder and see if the files occupying much of the space are things you don't need to keep, such as the swap files.

How would I identify "swap" files?

They'll be named swapfile0, swapfile1, etc. I would just drill down further into that folder and see where all the space is going, and have a look at the top few items in each of the folders to see where the large items are. Or just decide you aren't going to be a digital pack rat and delete the entire thing. What is the likelihood you're going to try to retrieve something from your old system at this point?