There are two schools of thought regarding the definitions of words. In one school, words have specific meanings that are stated in the dictionary, as given by the informal cabal that manages the English language (or, in other languages, an actual group such as the Académie française ). In the other school, words mean whatever people use them to mean.
It is perhaps a signal of the extreme controversy here that it is necessary to establish such a difference up front. Multiple sides are openly and explicitly engaging in linguistic warfare (note Chris Rufo bragging about having "frozen the brand"). We must thus provide (at least) two definitions.
Kimberlé Crenshaw is a prominent scholar often credited with coining the term Critical Race Theory in the 1980s.
Critical race theory (CRT) is an academic movement of civil-rights scholars and activists in the United States who seek to critically examine the law as it intersects with issues of race and to challenge mainstream liberal approaches to racial justice.
This definition is, unfortunately, completely lacking in describing the philosophical views of the theory's practitioners. Any attempt to ascribe specific details to Crenshaw CRT is highly controversial. A look at Twitter in June 2021 will find much shouting and little consensus regarding the topic of if and how "Intersectionality" is related to Crenshaw CRT.
The one conclusion I do feel comfortable in making is that Crenshaw CRT endorses the view that on-their-face racially neutral laws can have a race-sensitive disparate impact. This statement is not particularly new, controversial, or interesting.
Christopher Rufo, a prominent activist, defines critical race theory as an umbrella term for various "left-wing" race-related ideas, particularly unpopular and "crazy" ideas. He also has his own interpretations of who has these ideas and why they are popular; it is a logical error to allow him to "define" things in this way.
The definition above suffers from difficulty in determining if any specific idea is part of Rufo CRT or not. Some ideas are dismissed by "the left" as straw-men that nobody actually supports, and are posited simply to smear other ideas. In other situations, people with unpopular ideas try to dodge and weave to avoid being critiqued at all.
We can, however, simply look at those things Mr. Rufo himself has called out as being part of Rufo CRT. He says in the Atlantic that "critical race theory is an academic discipline that holds that the United States is a nation founded on white supremacy and oppression, and that these forces are still at the root of our society", which isn't much of a philosophy. In one post, he describes it as "promoting race essentialism, collective guilt, and neo-segregation", terms which are only slightly more clear in their meaning.
He repeatedly invokes the spectre of Marxism; I consider this a smear campaign rather than meaningful content as nobody knows what he means by Marxism - he certainly isn't referring to specific policies of Karl Marx . He does seem to be imagining his own "vast communist conspiracy", with echoes of Joseph McCarthy.
He also calls out straw-men (some of which are actual things that have happened) of terrible race-based policies; we will discuss those in the BTP definition.
Derrick Bell (1930-2011) was a professor of law at Harvard University who started using the phrase "critical race theory" in the 1970s. According to his Wikipedia article, he viewed Bell CRT as:
First, racism is ordinary, not aberrational. Second, white-over-color ascendancy serves important purposes, both psychic and material, for the dominant group. Third, ("social construction" thesis) race and races are products of social thought and relations. Fourth, dominant society racializes different minority groups at different times, in response to shifting needs such as the labor market. Fifth, "intersectionality and anti-essentialism" thesis. No person has a single, easily stated, unitary identity. Everyone has potentially conflicting, overlapping identities, loyalties, and allegiances. For example, person who has parents with different religious views, political views, ethnicity etc. Sixth, ("voice-of-color" thesis) because of different histories and experiences to those of white counterparts, matters that the white people are unlikely to know must be communicated to them by the racialized minorities.
Unlike the previous two definitions, this is at least a coherent theory. I have some issues with it; the implication (made explicit by many who support this view) that "there is never any biological basis to race" is a matter of experiment and not of faith. However, it appears to be a framework that explains multiple types of "race"-based issues (anti-Semitism in pre-war Germany, Hutu-Tutsi conflicts in Rwanda, etc.) and explains continued race-relation issues as not being based solely in the legacy of slavery or systemic issues.