The Name Game (Militant Edition)

Iraqis fleeing the fighting between the army and ISIS     Azad Lashkari/Reuters

Direct translations are often difficult, as we're seeing with the militant/terrorist group that is currently claiming top spot in the news these days. Are they ISIS or ISIL? What's the difference? And what is "the Levant," anyway?: http://www.bbc.com/news/world-middle-east-27994277