This takes place a few months after the end of THE WRATH OF KHAN, during which Spock sacrificed his life to save the crew of the Enterprise. Everyone is still feeling the repercussions of that, particularly Kirk. And McCoy is acting very strangely, especially when he breaks into Spock's quarters on the Enterprise and speaks in his voice. Sarek, Spock's father, visits Kirk and wants Spock's Katra, his soul, and is disappointed to find out that Kirk doesn't have it. But McCoy does-- Spock gave it to him at the end of the previous movie. While all this is happening Kirk's son, David Marcus, and Lt. Saavik, are exploring the newly created Genesis Planet. They come across Spock's coffin and find creatures growing from the bacteria. Soon, they come across a very young Vulcan child--and this proves to be a reborn Spock. Only he doesn't have any of his memories.
The war-like Klingons get wind of the Genesis device and blow up the ship that had dropped them off, stranding them on this unstable planet. Meanwhile, Kirk wants to return to the Genesis planet but is forbidden by Starfleet to do so. In fact, they are moth-balling the Enterprise because the ship is too old. So the crew manages to steal it and fly to Genesis, where there's a confrontation with the Klingons....
As STAR TREK movies go this is a good one, though the plot-line is completely written around undoing everything set up in the second movie. The one thing that always bothers me about this film is the casting of Christopher Lloyd as the evil Klingon adversary, Krug. I just kept on expecting him to go into his Reverend Jim character from TAXI, which diminishes any sort of threat he's supposed to convey.