Forbidden Planet is often described as a retelling of Shakespeare's The Tempest, and has a reputation as being one of the first "serious" science fiction movies. When compared to fare such as Earth vs. the Flying Saucers, or Them!, that's probably true, but I think contemporaries such as Destination Moon, This Island Earth, and of course The Day the Earth Stood Still, stand up well against it.
The plot is relatively simple. Earth space cruiser C-57D is on a mission to see what happened to an expedition to planet Altair IV, the spaceship Bellerophon, launched some 20 years earlier. They arrive to discover Dr. Morbius (played by Walter Pidgeon) alone on the planet, save for his daughter, Altaira (played by a scrumptious Anne Francis), and Robby the Robot, whom he "tinkered together" soon after arriving on the planet. The rest of the Belerephon crew were destroyed by a "planetary force" that mysteriously spared Morbius and his wife, now deceased of natural causes.
Eventually, Captain Adams (played by a still-a-leading-man Leslie Nielsen) and crew discover one of the secrets that Morbius has been hiding; a prehistoric super-race called the Krell who lived on Altair IV and built a nearly godlike civilization while the ancestors of humanity were still hiding from the dinosaurs. He shows them the "Krell wonders", and also the device that boosted Morbius's IQ to allow him to create Robby. The C-57D is attacked by an invisible monster that hearkens back to the original destruction of the Bellerophon, and there is a signature scene with the beast charging through the ship's force fields and being lit up by blaster fire.
Eventually, the truth comes out; that the Krell had completed a device that would transmit their very thoughts into action, and were destroyed in an orgy of destruction as their subconscious minds were given free reign. Morbius's subconscious has done the same, first destroying the Bellerophon after the crew elected to leave the planet that he loved, and later attacking the C-57D after his jealousy of his daughter was inflamed by the attentions of the crew to her. Morbius dies after setting the Krell machine to self-destruct, the monster is vanquished, and the crew (plus Altaira and Robby) departs.
One of the things that stands out about this film, aside from the outstanding acting, writing, and effects, is the use of humor, and the little touches of inter-personal relationships that don't really serve to advance the plot directly, but contribute to our sympathy with the characters, and thus our sense of involvement in the film as a whole. The scene where the cook gets Robby to produce gallons of bourbon, for example, and the early fumblings between Altaira and some of the crew. This was an iconic film in many different ways, visually as well as thematically, and set the stage for future films such as 2001: A Space Odyssey.