ISIS considers itself the "Islamic Caliphate" (a theological empire) and controls vast swathes of land in western Iraq and eastern Syria. They also have "allegiance" from different radical Islamic groups around the world (from Afghanistan to Nigeria) who "govern" self-proclaimed provinces.
Within the areas they control they have established a reign of terror second to none. They have institutionalized slavery and rape (particularly of adherents to the Yazidi religion who they view as devil worshippers) and have carried out genocide and ethnic cleansing of Christians, Alawites, and other Shiites and Yazidis in the territories they control.
They have struck with a vengeance beyond their territories. Suicide attacks in Baghdad, Beirut, and Ankara killed hundreds. In October 2015, they detonated a bomb aboard a Russian airliner leaving from Sharm el-Sheikh airport in Egypt, killing all 224 people on board. In November, they orchestrated a multi-suicide attack in Paris, killing 129 people. They have inspired "lone-wolf" terror attacks by sympathizers in places as far away as Ottawa and Sydney.
A bit of nomenclature:
You may have heard about ISIS referred to as IS, ISIL, or Daesh. All of these acronyms describe the group in question.
ISIS: Islamic State of Iraq and Syria was the name of the group when it captured Mosul in 2014 and became the terrorist juggernaut it is today. They named themselves that to assert their dominance in Syria (more on that later).
ISIL: Islamic State of Iraq and the Levant (aka Greater Syria) is the name that Obama uses to describe the group (pretty much only Obama uses it). Superficially speaking, it is just a translation thing.
IS: Islamic State is the name the group gave itself after a "rebranding" effort when they wanted to show off their global strategy (they wouldn't be limited to Syria and Iraq anymore).
Daesh: You may have heard French President Francois Hollande refer to the group by this name. This is essentially the Arabic acronym of the group. People assume that using this word somehow weakens them… it doesn't, because unfortunately in this case it is one of those "sticks and stones" things.
