Sartorial comes from the Latin sartor meaning tailor, and in the narrow sense it refers to tailoring and tailored clothes. (Merriam-Webster) However it is used more generally to refer to clothes in general, especially those with an understated elegance, in contrast to terms like "fashion" which refer to ephemeral styles. It is the sort of word directed towards educated, "in the know" audiences, not mass-market media.
Tailoring, as Wikipedia says, refers to a specific set of techniques and practices, generally involving making made-to-measure suits, coats, and similar clothes, using specific fabrics such as wool, woolen blends, and linen. There is a set of traditions and standard practices, many connected with Savile Row in London, the traditional centre of tailoring. Most people do not get made-to-measure clothes these days, and even if they do, there are various other expressions such as which are more precise than "sartorial" - a tailor is more likely to advertise as "tailoring", "made-to-measure", or "bespoke". So this technical sense of "sartorial" is not terribly common.
But sartorial also has a wider meaning related to clothes in general. Lexico has a good set of examples. Many of them use fixed phrases like "sartorial elegance" or "sartorial style"; sometimes you see the opposite, "sartorial nightmare/disaster".
"Sartorial elegance" in particular is a common phrase for a certain type of sophisticated, attractive dressing, that aims for classic good looks. The connotation is different to "fashionable", "trendy", or "stylish", which imply adherence to the changeable rules of fashion. Classic style denotes clothes that are understated and well made, and so will last and will remain stylish. (Here's a fairly typical example of a magazine telling its readers how to dress in classic style; such guides are regularly published and the notion of what is classic changes gradually).
The main usage seems to be in elite journalism and fiction (rather than mass-market publications), where a more unusual and elegant-sounding word denotes class, style, and elegance. It is also often used to refer to people who fail to meet up to standards: "sartorial disaster", "sartorial faux pas"; here again the word denotes certain standard, that in these cases is not being met.
There is no gender divide in the word, based on Lexico's examples. Both men and women can dress in classic style, and be praised for their "sartorial elegance" or "sartorial style". The latter phrase is distinct from "fashion sense" which implies more of a knowledge of what's trendy; "dress sense" is similar to "sartorial style" but I get the feeling that "sartorial" is used because it sounds more sophisticated, and hence relates to being better-dressed. "Sartorial" is a less common word than "fashion" and thus requires a more elegant audience.
Typical usage would be:
- "Such-and-such a celebrity is known for their sartorial elegance." Used in something like Vanity Fair or the New York Times rather than a tabloid.
- "Such-and-such a person (often the author) made a sartorial error/mistake/slip-up." Used in humorous journalism in a similarly sophisticated publication, or in middle-class fiction, to indicate that someone who should know better has worn the wrong thing.
You wouldn't use it to talk about clothes with no connotation of elegance; or for "fast fashion" that will be in the bin next year.