ISIL subscribes to a radicalized Sunni version of Islam. Everyone who lives within an area subject to ISIL' jurisdiction is subject to ISIL' extreme version of Sharia, Islamic law. Muslims who do not practice ISIL' version of Islam are apostates; everyone else is an infidel. According to ISIL' version of Sharia, it is the duty of faithful Muslims to kill both apostates and infidels. ISIL' numerous beheadings reflect these beliefs.
So far, ISIL appears to be a successful insurgency and not a terrorist organization. Terror organizations commit violent attacks against innocent people to achieve political gains. ISIL rarely if ever does this. Their violence is very different from the "performance violence" of a genuine terror group. I have found no public evidence that ISIL has credibly threatened Europeans or US citizens at home. Most importantly, terror organizations do not conquer and then govern territory.
ISIL' heinous acts (e.g., beheading hundreds of people including two American journalists) rarely have demands attached. When ISIL does make demands – whether political or economic – those demands represent a form of extortion or kidnapping, not non-state terrorism as defined by experts like Harvard's Louise Richardson.
Correctly identifying ISIL as an insurgency instead of a terror organization is an essential first step in addressing the problem that ISIL poses. First, recognizing that ISIL is not a terror organization and does not directly threaten Europe or the United States means that immediate action is unnecessary. Precipitously acting to defeat ISIL would require inserting ground troops, conquering territory, and then ruling that area until a new government (or existing ineffectual governments) can assume the tasks of governance. Failure to provide interim governance would lead to a repeat of the chaos that occurred in Afghanistan and then Iraq following the US conquests in 2001 and 2003 respectively.
Second, sending more arms to Iraq is not the answer. Iraq is already one of the most heavily armed states in the world. If Iraq had fewer weapons, ISIL would have had a much more difficult time arming itself.
Third, sending more arms to Syrian rebels is also probably not the answer. The US has provided training and equipment to some of the rebels, but longer-term loyalty of those rebels is always in doubt. Many Syrian rebels hold Islamist views close to, or sympathetic with, ISIL' version of radical Islam. Like ISIL, the rebels are opposed to Assad's regime for political, religious, and other reasons.
Fourth, aiding the Kurds in their battle against ISIL may have long-term unwanted consequences. The Kurds seek their own nation, which would incorporate parts of Iraq, Turkey, etc. Supporting the Kurds in this conflict will better position them to achieve that goal, adversely affecting nations that lose people and area to the new Kurdistan. Supporting the Kurds may also be the death knell of Iraq as a unified country.
Fifth, the underlying issue is that the Middle East now consists of states whose borders European colonial powers and the US established, borders that often have little basis in history, geography, or population. The unraveling of those borders and the continuing struggle of people in that part of the world for self-determination (which is not necessarily synonymous with democracy) is not a problem that the US, NATO, the UN, or any other external coalition can solve.
Sixth, politicians (e.g., President Obama in his speech to the US on 2014 eve of 9/11) find it convenient to describe ISIL as a terrorist organization. The term evokes a visceral response from hearers, a response that condemns the group and calls for action. This usage of the term terrorist organization, by stretching the term to include very different types of threats (insurgency and terror threats) conflates very different types of problems that require very different approaches.
Much resentment toward the West exists throughout the Middle East. The West for almost a century unilaterally exploited and benefited from the Middle East's vast reserves of petroleum and then left a tragic legacy of colonial imperialism. Full of guilt over the Holocaust (which was an inexcusable moral failure by the US and European nations), the US and Western Europe used the umbrella of a nascent United Nations to impose the modern state of Israel on Palestine. Nowhere else in the world would a 2000 year old claim to land based on religious writings and from which the claimants had largely been absent for most of that time, trump people in actual possession of the land.
Consequently, ISIL is not a problem that the US acting alone or in concert with other Western nations can solve. The people, leaders, and states ISIL directly and immediately challenges must take the lead and primary responsibility for fighting against ISIL. Other nations can provide limited military assistance (e.g., some airpower or resupplies of munitions), but the fight must be fought and won by the people that ISIL would rule.
Logically and reasonably, if ISIL can fight mostly with weapons already in the area using mostly local recruits that ISIL trains and leads, then ISIL' more numerous opponents, if equally motivated, should be able to prevail with fighters they recruit, train, equip, and lead. Arguing that ISIL' opponents need Western training, leadership, or assistance reflects both Western hubris and an incorrect disdain for Arabs as inferior warriors and leaders.