Understanding NASA's Mission Control Map

We see lots of video from mission control during ISS operations and at the front of the room there are big screens displaying things which are important to all the controllers. One of the most prominent displays is the map which shows the location of the ISS & other spacecraft as well as a bunch of cryptic, wavy lines.
So, let's explain what all these mean and why they're important to the control room staff when operating the most complicated machine ever flown.

Image Credits:
NASA
Roscosmos
James Blair
Bill Ingalis