What kind of window is this (and how do I fix it)?
I had never seen these before, but I was finally able to track down some information and replace them.  My windows bear a "Viking" name, but I didn't find anyone selling parts under that name.

However, the useful term for these are "channel balance" or "block and tackle balance" windows.  With that term, it's possible to find replacement parts and even some youtube videos showing the process of replacement.