Generally, words like "extremism" are pejorative and there really is no substance. However, there are some general categories of definitions in political science.
I had the fortune to meet Dr. Haider-Markel while at the University of Kansas. This section is taken from my notes for a graduate course in political extremism and terrorism he taught in 2011.
What is Extremism?
Within political science, there are two general approaches to the idea of terrorism: people are either extremists because of their opinions or because of their (in)ability to flex on those opinions.
The first view is that opinions themselves can be extreme, and so a person who has those extreme opinions is an extremist. This definition tends to fall apart fairly quickly. How do you know that an idea is extreme? Usually because it is different than the speaker's opinion. Some people would say it's because an idea is statistically unusual, however having an unusual opinion is not a strong predictor of engaging in extremist activity (like blowing up a building or hijacking a flight, etc.).
The second definition is somewhat more robust: a person isn't an extremist because of their belief, but because of how they maintain it. They are unwilling to compromise or change their beliefs, even in the face of evidence or reason. They are passionate about their beliefs, even in times when it is otherwise socially unacceptable. Anyone with a different view is the enemy. Their ideology explains everything of importance with no possibility of being wrong.
The question uses Nazis as an example, so I will jump the shark and start with Nazis (this content is an application of the ideas above and not directly from Dr. Haider-Markel). Nothing about the content of the Nazis belief was particularly extreme: eugenics, forced sterilization, social Darwinism, anti-Semitism, and many more of their beliefs were all normal (although obviously not shared by 100% of the world) at their time. Many rational people could buy into any of these positions: a person might support eugenics for an infinite number of reasons. Many Nazis were not extreme in-and-of-themselves - they could be normal people with normal beliefs and entirely functional in society.
The course materials included a Youtube video that I can't locate. It was a man in a city council meeting (in the US) angrily supporting some view. When questioned with some fairly innocuous questions, his face literally turned red and he started shaking from the rage of having his viewpoint questioned. That is an extremist.
Is that it? What about violence?
We often hear about extremism in the news through an association to political violence. Extremists are great candidates for radicalization. Radicalization is social and educational process that makes a person capable of extreme actions to support their views.
A great example of radicalization is the Weather Underground, a terrorist group active in America in the 1960s-1970s. The group was composed of Ivy League students. They didn't have any experience with crime or violence. However, they were led by extremists: people absolutely dedicated to ending the United States' military involvement in Vietnam.
Because of their strong opinions, they could be radicalized. They became ingrained with others with similar opinions and isolated themselves from people who didn't agree (effectively amplifying their world view and conviction). They learned to build bombs, use guns, incite riots, smuggle, and many other things. Eventually they put their plans into action.
So extremism is linked to political violence. It's a kind of conditioning factor.
Is it a left or right issue?
No. There are extremists on both the left and right - if the left and right exist! There is no certain kind of belief that is extreme (see the first section, "What is Extremism?"), so there is no connection between the "left" or "right" and extremism.
Additionally, the terms "left" and "right" are imprecise for any rigorous thinking. They are a kind of social shorthand that is really only useful for talking to people who generally understand the world the same as you.
However, because you asked, here are some examples people generally consider to be far-left extremists: