There are two definitions of extremism in common use. Because we seem to be getting more extremists than ever before, I want to come up with a useful definition.
The first meaning is "outside the norm of a given society", which is used in the Wikipedia entry on extremism. I don't think this definition is right any more. Maybe it used to be accurate, when you could assume that the norm of any society was moderation. But if the norm in a society becomes extremism, then is being outside the norm still extremism?
The second meaning is "Uncompromising". This, I think, is a more generally applicable definition. For one thing, it gives a universal standard of behaviour to judge extremism by. If someone or some government is willing to compromise, it is not extremist. According to this definition, it is possible to be a moderate minority within an extremist majority.
There are examples of extremism throughout history, using the second definition. The Spanish Inquisition would be extremist because they did not compromise with heretics. Instead of compromise, there was burning at the stake and torture. The Nazis were extremists, as shown by the Czech Peace Agreement. The British and French signed a compromise deal for peace, agreeing to give Czechoslovakia to the Nazis in return for peace without further land claims. Instead, the Nazis invaded Poland, which clearly established them as extremists. No real compromises could be made with the Nazis or Hitler, despite their willingness to talk of peace. Ultimately, this revelation of extremism made it impossible for Britain to surrender on good terms after the fall of France, even though Hitler pleaded that he meant them no harm, and would allow them to keep their sovereignty.
The Nazis went on to show themselves as extremists in many other ways, but were finally beaten by a coalition of moderates.
There are good examples of extremism today in certain religions. Extremist Moslems want to wipe Israel off the map, and the USA too, if they could. Extremist Jews want all Palestinian disputed territories for themselves, and want to deny independence to the Palestinians, and would resort to any means to accomplish that including killing. I suppose you could say Jesus was an extremist, but an extremist for peace. Now, extremist Christians want to bomb Iran, and kill doctors who perform abortions.
None of these religious groups is interested in any compromise. All would prefer the world to end than give up their demands - and that gives us yet another way to define extremism. If you would be willing to kill yourself and the rest of humanity rather than accept a compromise with another, you are an extremist.
Most people are not born extremists, but are sometimes driven to extremism by forces beyond their awareness. First, a very fearful life with a lot of uncertainty can make one predisposed to extremism. Certain powers like to promote fear and extreme views in order to benefit from the support of extremists.
If you were talking to an extremist, whether a friend or family member, or a total stranger, is there some way to know this is an extremist? You you use the definition of "outside the norm", you may not be able to ever tell. But if you use the definition of "uncompromising", it is easier to figure out if you are dealing with a moderate or an extremist. One clue is that an extremist will hardly ever admit their side can be wrong. But for the most reliable indicator of extremism, try proposing a compromise and see the reaction you get. An angry reply, a refusal to discuss further, or both, may be a clue you are talking to an extremist.
There is not much point in discussing things with extremists. While you may learn something of their point of view, you will simply become frustrated by their means of discussion. Instead of logically addressing an issue, you will find they present you with multiple arguments, usually handed on from other sources. If you question them about the issue on an obvious flaw of logic, they can quickly change the subject. Most of their opinions are designed not for truth seeking but for provoking anger and confrontational arguments.
Picture: from the Anti-Defamation League website discussing extremism.
The ADL was set up in 1913 to combat anti-Jewish extremism Their website names a lot of extremist people, organizations and movements. I could not find a mention of the Jewish Settler movement, although according to my definition, those people would also be extremists because they are not interested in compromise. On the other hand, maybe the mandate of the Anti Defamation League is only to attack non-Jewish extremism. Would that make the ADL itself an extremist organization?
This was the 1913 incident that gave the ADL its start (and re-started the Ku Klux Klan).