The starship Reliant (no word on whether the captain's chair has rich Corinthian leather) comes by and doesn't notice that one of our planets is missing. Long story short, Khan et al. seize control of Reliant and use it to try to steal something called the Genesis Device, which can "create life from lifelessness" on an epic scale and which, of course, can also be perverted into a terrible weapon, also on an epic scale.
Meanwhile, back at Starfleet Command, Admiral Kirk is feeling old and, lugubriously, has drinks with McCoy; Enterprise is under command of Capt. Spock and is being used as a training ship; Kirstie Alley is introduced as Spock's protege, the half Vulcan, half Romulan Lt. Saavik; and we are introduced to the kick-ass Kobayashi Maru no-win training scenario.
As Enterprise starts out on a training cruise, Admiral Kirk gets a call from the head of Project Genesis (the two, apparently, used to be an item) and wants to know why Reliant is on its way to pick up the Device. So Kirk takes command of Enterprise again and leads the ship to intercept Reliant, whereupon he shows precisely why he was promoted up and out of the way when he nearly gets the Enterprise destroyed (To be fair, Spock contributes to this idiocy as well, but then, he too, is no longer in command of a ship of the line, either). Eventually, however, Kirk and Spock use their knowledge of the Reliant's ATM PIN code to shut down its shields and fire back.
Stuff happens, and eventually Enterprise leads Reliant on a wild goose chase through the Mutara Nebula, where neither ship will have effective shields or sensors. Enterprise cripples Reliant and kills all of Khan's people. As a last act of vengeance,
Oh, no. What to do?
Spock leaves the bridge and heads down into engineering where he does something in the highly irradiated dilithium chamber that fixes the warp drive. In the process [SPOILER] he dies. (It's never been entirely clear why they don't have robots in the 23rd century to handle things like this (or why one of the engineers who's wearing an actual radiation suit didn't go in there); of course, back at Starfleet Command, they use a guy with a mop and bucket to clean the floors, but I digress). And Enterprise gets away.
After some poignant words and bag-piping, they shoot Spock's body out a torpedo tube onto the newly formed "Genesis Planet." And we all go home.
Wrath of Khan is by far the best (and my favorite) of the Trek movies, notwithstanding certain sparks of idiocy and over-acting. (Kirk does redeem himself with several awesome moments, not least of which is his solution to Kobayashi Maru). The Kirk-Spock-McCoy dynamic is fun, and Saavik is an effective addition. As for Kirk's son, well, the less said the better. (And, yes, this one came out a lot snarkier than I'd initially intended, but these things have bothered me for 25 years...).