I was wondering what books about English grammar are:
- the standard ones written for linguists or linguistics students, and
- the standard ones written for people who are neither linguists nor linguistics students?
For linguistics-oriented texts, most people here seem to be fans of Huddleston & Pullum (2002), McCawley (2nd ed., 1998), or Quirk et al. (1985). Expect a vigorous debate in the comments section about which is best.
I'm currently reading Huddleston & Pullum; I'm not a linguist-in-training, just a nerd with too much free time. I'm not sure if you're even meant to read it in its entirety; it's about 1800 pages and quite dense. Annoyingly, the Kindle edition isn't formatted properly for phone screens, so you have to keep zooming in and out. But it is interesting, if you enjoy extremely thorough discussions of seemingly minor details of syntax.
As for people uninterested in linguistics, it will depend on your goals. If you're looking to (say) become a more eloquent writer, I don't think grammar textbooks will help you much per se.
@alphabet's answer covers professional grammars. The problem with the other category ("people who are neither linguists nor linguistics students") is that there are no "standard" grammar texts for them. That's like asking what is the standard text on vector calculus for people who have no prior math beyond multiplication tables and long division.
Anglophone education (excluding ESOL) does not teach students anything about the sounds of the language, nor its actual grammar and usage, because the systems all assume their students are native speakers, and therefore don't have to study that. Far more effort is devoted to "mastering" the complex English spelling system than to the equally complex sound system.
As for grammar, pseudo-Latin gobbledegook predominates, and teachers whose own teachers have never learned Latin make up their own paraphrases and create their own educational theories. All of this has to be left behind if you really want a book that teaches you something, so in fact you hafta study some linguistics just to be able to follow anything that tells the truth about English grammar, because Anglophone students are usually educationally deprived. Not your fault, of course.
For that purpose, I recommend either of David Crystal's Cambridge Encyclopedias: of Language, or of the English Language. Or both; there's little overlap. Both have good glossaries and references and cover everything, and they're cheap and fun to read.