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.