The IMDB summary describes it best:
Five individuals from five nations, including the "Superpowers," USA, USSR, and China, suddenly find themselves on an alien spacecraft. An alien gives each a container holding capsules. No power on earth can open a given container except a mental command from the person to whom it is given. Each person has been provided with the power of life and death. Any of these individuals has the capability to instantaneously launch the capsules to whatever coordinates he/she chooses, and each capsule will then eradicate all human life within a 3,000-mile radius of its designated location.
Since it's 50s sci-fi, most of it is not very subtle. Here are the folks who've been chosen. Can you spot the Communist?
And why the hell would the Alien do this? Because his people's planet is dying and he wants to move them onto the Earth, but his personal code prevents him from wiping out humanity himself, so he's giving them the opportunity to do so. If they don't, I guess he'll just go away and play his little game with another planet, huh? The name of the film, by the way, comes from the Alien's confidence that we'll wipe ourselves out within twenty-seven days.
There's really nothing else to this film that you can't guess for yourself. The Chinese lady kills herself immediately because she can't handle the responsibility, the Englishwoman throws her capsules into the ocean, not wanting any part of any of this. The German scientist gets hit by a car. The Russian soldier is interrogated by the Kremlin because they simply must have this power for themselves. The American and the Brit, of course, go to the US authorities who are wise and decent - but then they can't resist testing one of the capsules out, either.
Eventually, the German scientist notices there is writing on the capsules. Nobody else ever actually looked at theirs, of course, but it turns out that the capsules only kill enemies of freedom and the capsules are all set off, destroying all those nasty little bad guys who hate freedom. You know who I mean as well as the audience would have.
Then, because there's all this empty land on another continent, the United Nations invites The Alien and his people to live there. How nice.
Unlike this movie.