The episode that Stanley Kubrick stole his most important ideas from for 2001: A Space Odyssey. Of course, that's not exactly true. The idea of an artificially intelligent computer becoming a problem popped up in science-fiction at least shortly after Alan Turing re-popularized the idea of artificial intelligence in the 1950s via what's become known as the "Turing Test" for just that property. Also, Kubrick's 2001, written in conjunction with sci-fi novelist Arthur C. Clarke, began production in 1965, and there are more ideas there than just AI gone haywire, as there are in The Ultimate Computer. But this episode underscores that Star Trek deserves consideration as "serious artwork", consideration that it doesn't often receive outside of the Trekkie community. Even though Star Trek didn't likely influence 2001, the reverse isn't the case, either; rather, both works arrived at similar ideas due to mindfulness towards relatively cutting edge ideas in science and science fiction.
By this point, in case you're looking for a plot summary, you surely know that The Ultimate Computer has something to do with an artificially intelligent computer. It arrives on board the Enterprise courtesy of Dr. Richard Daystrom (William Marshall, soon after The Ultimate Computer to forever be best known as Blacula), a computer genius who long ago designed the basics of the system currently employed on the Enterprise. To test his new system, which is supposed to be able to run the ship more or less by itself, Starfleet orders all but 20 crew members off of the Enterprise and organizes a fairly elaborate war game scenario. Of course, we know as soon as we find out the premise that it's probably a recipe for some kind of disaster.
Aside from the usual AI kinda themes, writers Gene Roddenberry, Laurence N. Wolfe and D.C. Fontana use the episode for a nice exploration of ill-conceived idealism, more general technological skepticism and unease, overly fervent parental apologetics, and difficult utilitarian ethical decisions. The performances are excellent as always (and I always wished that Marshall would have had a more prolific career), and we get a bonus treat of a very Kirk-like head of another Starfleet ship, Commodore Robert Wesley (Barry Russo).