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Why is a professorship sometimes called a chair? Are chairs only the correct term when the professorship has been endowed? (I don't think so, because endowed professorships are rare in the UK, but people still speak about chairs).

What's the origin and correct usage of the term?

Edit: Thanks for the responses, I would like to slightly expand on the 'usage' part of the question by asking whether every professor at every university could correctly be said to hold a chair? If it varies, what determines who has a chair and who is just a professor? Perhaps the term is more common in the US because the term professor is used more widely than in the UK, for relatively junior faculty (who would be called lecturers in the UK).

etymonline.com:

chair (n.)

"a seat with a back, intended for one person," early 13c., chaere, from Old French chaiere "chair, seat, throne" ... from Latin cathedra "seat" (see cathedral).

Figurative sense of "seat of office or authority" c. 1300 originally was of bishops and professors. Meaning "office of a professor" (1816) is extended from the seat from which a professor lectures (mid-15c.). ...

The etymology is much more obvious in Romance languages: see e.g. the Spanish word catedrático.

Note that there's a specific type of chair which is also used figuratively to refer to the authority of the person who is entitled to sit in it: throne. Usage is probably parallel, although digging out examples to demonstrate this would be a non-trivial research project.

I see from comments that there are two opinions about what the question meant by asking about usage. To address the other opinion, in as much as it's not already answered by user2768: professor in British English is reserved to those who hold a chair, which may be an endowed or a personal chair. In American English it is used far more widely. Cognate words in other languages may be used even more widely still: in Spanish, profesor is used for teachers from primary to tertiary education, although as mentioned above there's also a word catedrático which corresponds to the British professor. I would expect that most of the Commonwealth follows the British usage, with the possible exception of Canada, but I would not be surprised to be corrected on this in comments.

Professors hold a “chair” in a subject which can be either established or personal. Established chairs exist independently of the person who holds it, and if they leave the chair can be filled by someone else. A personal chair is awarded to a specific individual in recognition of high levels of achievement. If they leave, there is no guarantee the chair will be available for someone else.

Source: https://academicpositions.com/career-advice/uk-academic-job-titles-explained

It's a figure of speech called metonymy. It's like referring to the UK government as "Whitehall", or to the monarch as "the Crown".

It makes a useful distinction between the job/office, and the current individual job holder.

I think that any professorship can be referred to as a chair.

My mother is a full professor, but was only titled 'chair' once she became the 'chair' of the English department. In that context, it connoted her as the senior-most professor in her department, with official administrative duties, including hiring decisions for the department, being a part of tenure award decisions, etc.

Also, other professors I've known have held chairs with someone's name attached, and this was an honorific rather than an administrative position. I don't think it's accurate to say any given professor, even any full professor, is considered 'chaired' by default.