Traditionally and historically (at least in the past 100 years or so), a socio-political "conservative" was someone who had a strong stance towards retaining (or going back to) traditional societal norms and values, generally of very prudish nature.
Conservatives often acted as self-appointed "moral police" of society, protesting, acting against, and introducing legislation against anything that they considered immoral, lewd, or detrimental to society (especially the youth). They were very much pro-censorship, and banning anything that they deemed inappropriate or dangerous. They had very traditional, strict, and usually very religious views on romantic relationships and marriage, and wanted to exert great amount of control over people's sex life (eg. sex before or outside marriage is fornication and adultery, and if not illegal, it either should be, or at the very least it should be very shunned and frowned upon.) Quite often, conservative people promoted strict social norms in social interactions, policing speech and enacting policies on what can be said and how.
Historically, ultra-conservative people were often also genuinely racist, having the strong belief that race plays, and should play, a role in societal class and standing, sometimes even having the belief that people of "inferior races" were actually incapable of the same intellectual level as the people of the "superior races" (the more back to the past we go, the more prevalent these beliefs were.) They often thought that different rules should apply to different races, and likewise often believed in keeping the races separate. The so-called Jim Crow laws were a clear manifestation of this mentality.
In contrast, more "progressive", or "liberal", people sought for a more equal world, with more freedoms and less restrictions on people's rights. It's not anybody's business who has sex with whom or when, people shouldn't be forced or pressured into marriage if they don't want to, people's rights to free speech and free expression should be guaranteed, almost all censorship should be removed, people (especially women) should be more free to dress and act as they like without repercussion (as long as they aren't hurting anybody), and people should most definitely be considered equal before the law, and before society. Racial discrimination and segregation was deemed especially abhorrent, and a crime against humanity. All people should be considered equal, and of equal value, completely regardless of sex and race.
Today, somehow, it seems that these roles have almost completely reversed, especially when it comes to the conservatives and "progressives" in the United States, but also so in most other western countries.
Today it's the conservatives who are strongly defending the right to free unrestricted speech, of anybody having the right to express their opinions, no matter what it is, without legal repercussion. They are the ones who support and promote the notion that people should be treated and judged equally regardless of sex and race. They are the ones supporting universal human rights. They are the ones who, in increasing manners, are opposing any sort of racial segregation and prejudice against people based on sex and race. In a significant part, they have even embraced homosexuality as an acceptable way of life, and consider them to have the same rights as anybody else. In fact, some of the most vocal and prominent conservative public personalities are openly and unashamedly homosexual, and fully embraced by most conservatives (Milo Yiannopoulos being the most famous example.) Perhaps not universally so, but in a very significant majority.
The "progressives", on the other hand, have reversed their attitudes in so many ways that the moniker "regressive" is well earned for them. Rather than promote and stand for universal unrestricted free speech, they want heavy restrictions on it, and heavy censorship. They want people punished and silenced for expressing opinions that they don't like. They want to limit and restrict people's freedom of peaceful congregation. They will even physically attack people that express opinions they don't like.
They want women in media to be depicted in a less revealing and sexualized manner, for them to be clothed more prudishly and conservatively (the irony of which completely escapes the "progressives"). In fact, they want women who are in jobs that depicts them as sexualized to be fired (and have already succeeded in it in several instances). They want censorship in media, both in respect to "lewd" depictions of women, as well as in regards to the "wrong" opinions. They will also meddle with people's sex lives, and shun people who express the "wrong" preferences.
"Progressives" have taken racist attitudes to a whole new level. They openly and blatantly divide people by race, and assign rights, privileges, blame and guilt onto them based on which racial group they belong to. They are openly and blatantly racist against certain groups of people based solely on their race, often deriding all of them for that sole reason. Many of them want full-on racial segregation.
This is a complete bizarro world, where everything is reversed. Conservatives have become progressive, and progressives have become conservative.