It's all Christopher Nolan, folks. Director, Producer, Writer. It's famously incomprehensible, and I'm at that age where I don't comprehend the incomprehensible easily.
I got it, though. Mostly. I'd probably get it entirely after rewatching it five or six times, and reading a bunch of explanatory know-it-alls on the web.
Here's the deal (and stop reading if you don't want things spoiled): bad guys from the future are looking to destroy our world, so they've provided time-travel tech to a present-day bad guy, Sator. Who is, in return, attempting to reassemble … something nasty, an algorithm that will aid in humanity's destruction.
The gimmick with the time-travel thing: going back in time works, but you continue moving backward in time when you disembark. (There's a handwave to Feynman's wacky idea that a positron is just an electron moving backward in time.) And of course, the rest of the world appears to be moving backward to you. This gives rise to some pretty neat special effects.
All this is (eventually) revealed to the protagonist, helpfully named "Protagonist". His goal is to defeat Sator's scheme, and hopefully also save Sator's beautiful wife and cute son in the process.
So that's it. Now to look for some websites to tell me how mistaken I am in my analysis.