What is the Alt-Left and how different are they from the Far-Left or Ultra-Left? Are they similar to the Alt-Right in any way?
The alt-left is not really a thing unless you live in Switzerland, where it is a political party.
In the US, it's a term that got cooked up by the alt-right to describe their fantasized progressive equivalent. The problem, of course, is that there is no such thing.
There's the oddball violent antifa, but where the aim of such would-be alt-left might be fighting intolerance or a better (re)distribution of wealth, that of the alt-right is white supremacy. Personally, I find it hard to accept that the two are equivalent. The alt-left is leaning far left, whereas the alt-right stands for everything the US fought against during the Civil War and World War II.
The expression's history has a few interesting tidbits. Long story short, it seems to have appeared in Reddit alt-right groups. There is about zero reference to it in Google (except for the keyboard shortcut) prior to the last US presidential campaign. It got some uptake after a Vanity Fair article denouncing it as a problem. (There also were a few calls to embrace the term after the article.) It got much more serious uptake after Trump used the term in his "on both sides" commentary after Charlottesville.
There is no "alt-left". It's a term made up by the right to far-right in an attempt to create a false equivalence between white supremacists and those opposing white supremacism:
It's a "made-up term" used by people on the right to "suggest there is a similar movement on the left," [Oren Segal, director of the Anti-Defamation League's Center on Extremism,] said.
But there's no equivalent with the anti-Semitic and bigoted groups that call themselves "alt-right", he said.
[George Hawley: ] "There is no such movement as the alt-left. Obviously, there are left-wing extremists but there is no congruence between the far-left and the alt-right."CNN: What's the 'alt-left'? Experts say it's a 'made-up term'
Researchers who study extremist groups in the United States say there is no such thing as the “alt-left.” Mark Pitcavage, an analyst at the Anti-Defamation League, said the word had been made up to create a false equivalence between the far right and “anything vaguely left-seeming that they didn’t like.”New York Times: Alt-Right, Alt-Left, Antifa: A Glossary of Extremist Language
Ultimately, the intent seems to be to frame alt-left as the opposite of alt-right and create a false equivalence between groups on the far ends of the right and left. But here's the thing: No left-wing group has ever called itself the alt-left. And the groups smeared by the alt-left label don't include anything like the heinousness of overt white supremacism that has increasingly defined the alt-right.
It's a blanket term some right-wing media commentators and white nationalists have taken to throwing over groups they disagree with Wired: There Is No 'Alt-Left,' No Matter What Trump Says
"Alt-right" on the other hand is a self-applied term that was made up by white supremacists themselves for propaganda purposes.
There are of course far-left ideologies such as various sorts of communism, anarchism, etc. They are not comparable to the so called "alt-right" though; while some may contain racism or antisemitism, it is not central to any left-wing ideology, while they are central elements of the so called "alt-right". There are of course a lot of other differences, but that seems to be outside of the scope of this question.
The alt-left, regressive left, or illiberal left are terms used to describe a subset of people on the left. The three terms may not be exactly the same, however they are close enough that I felt it makes sense to talk about them together. These terms seem to collectively amount to a certain type of criticism of some aspects of the left. First, it should be noted that the expressions are not used as self-identification, but rather used by opponents of their beliefs/actions.
The people in question are often committed to some degree of the blank slate or egalitarianism. From this it follows that all/most differences between human groups are caused by differential treatment of those groups, or by differential media portrayals of group members [*]. Particular areas of controversy are: sex differences, ethnic differences, class differences, cultural or religious differences. The main offenders are generally the patriarchy, white-supremacist society, the capitalist society, etc.
The terms are generally not used just because of their beliefs, but for their actions against people that challenge those beliefs.
When Lawrence Summers or James Damore suggested that there may be biological differences between the sexes in terms of e.g. interests that lead to different statistical representations of the sexes in various fields, then that goes against the beliefs of the "alt-left". But it's not just that they disagreed: they had to be fired, and it doesn't matter if they presented evidence. There was no reason to discuss the evidence, it was simply sexist stereotypes. Despite several experts generally agreed, and it is not something suggested without evidence. Instead of reasonably discussing the evidence, they are labelled (neuro-)sexists.
Another area of controversy is on the subject of Islam. Instead of reasonably discussing the topic with critics of Islam, they label them Islamophobes and/or racists. This is why Maajid Nawaz -- a practicing moderate muslim himself -- coined the term "regressive left". He believes that, while there are right-wing groups who are too eager to stereotype Islam/muslims, there are also many left-wing people who are too reluctant to criticize Islam. This is a sentiment shared by some other muslims and ex-muslims.
It should be noted that just because there are people who are too quick to call someone sexist, islamophobic, or whatever, that does not mean that the terms "sexist" or "islamophobe" are always misused.
These are just a few examples, I could easily give more areas in which such controversy have arisen. Generally, the "alt-left" view their opponents as so evil (sexist, racist, islamophobic, etc) that they don't even deserve to have a rational conversation about the topic at hand. So it is not just disagreement, but the desire to shut them down. Thus we see trends of disinvitations of university campus speakers, and in the most extreme cases riots and violence (see e.g. Antifa).
In conclusion, "alt-left" is one of several terms used to describe a certain faction on the political left. Certain beliefs are typical of the "alt-left", however it is not just their beliefs, but rather their desire to shut down people who disagree them with them, or their inability to honestly discuss an issue, which -- at least according to some people -- justify the name.
Many people on both sides falsely label their opponents.
The fact is there are bad on both sides and the simplest answer to the stated question is
Alt-left is the name given by conservative groups to the most extreme members of the left
Just as alt-right is used to describe the extreme elements of the right. (including, as state in other answers, by those extreme-right wingers themselves). So there is no real difference in labeling them far-left or alt-left.
Most commonly it is used as a term to denounce groups and individuals committing violent crimes (notably antifa and various individuals on university campuses) and those who use fraud to progress their agenda (fake hate crimes) and who push for false equality and affirmative action (I'm more equal than you)
To my ears, alt-left/far-left/ultra-left is the same. Ditto alt-right/far-right/ultra-right.
It describes zealots who are irrationally convinced in the ultimate truth of their politically beliefs, find everybody who even mildly disagrees evil, intolerable to free expression of political thought that doesn't coincide with theirs, and willing to use violence to promote their political agenda. Basically extremists, in both their views and their method.
Both left and right had multiple examples of that throughout the 20th century, and, I'm afraid, both are present in US now. Much of 2020 was a bacchanaly of alt-left, with multiple cities burning by self-righteous leftist zealots, and January 6 2021 alt-right did a similar thing in DC.