That's a type of Windows executable.
I believe those tools support Mac OS X because someone took the time to port them.
Has anyone had any experience with Fink?
Fink is not an option without having the source AFAIK.
EDIT: I think I can answer a few questions from this thread.
It is. At least, all modern Macs are.
I've used WINE (WINE Is Not an Emulator) a lot in the past. You got the explanation of what it does wrong; it's supposed to be able to run ANY Windows executable. It's a compatibility layer for -Nix OSes. What this exactly means is lost to me, but from what I've read it's similar in practice (not in terms of code) to how Windows 10 runs Windows XP programs.
In terms of what to use, I would suggest either WineBottler or Crossover. I'm more familiar with WineBottler, as Crossover costs money. I've also tried WineSkin in the past, but I can't recall what kind of successes I had with it.
In terms of things going wrong, you might get anything from graphical glitches to it just not working. I would still highly recommend trying it if nothing else works. Be sure to install the correct Winetricks to your bottle and check the error logs to see what Winetricks you may have missed.