I'd put it in a list of both Top 25 science-fiction movie and Top 25 horror movies ever made. And the term it made popular 60 years ago -- "pod people" -- remains in our mass-cultural lexicon to this day, used primarily now by people who probably have never seen the movie, much less read the novel it's based on.
Made on a shoestring budget, Invasion of the Body Snatchers became a surprise horror hit in 1956. Don Siegel's direction and Daniel Mainwaring's script keep things tight, perhaps a bit too tight when it comes to the rapid acceptance by several characters of an invasion of pod people. But that's a minor quibble.
Kevin McCarthy does a fine job portraying the gradually mounting paranoid exhaustion of a man who doesn't dare go to sleep, and Dana Wynter is fine as well as McCarthy's love interest.
It's the creepiness of the concept, and that concept's portrayal, that makes the whole movie sing. You will be replaced by an emotionless replica of yourself -- and that replica will talk about how great this development is. This first adaptation keeps the mechanics of the 'changeover' murky, which is a plus. A couple of later adaptations would make the switch from person to pod-person a piece of graphic visual horror (and a job for the garbage-men).
The studio found the original version of Invasion of the Body Snatchers too disturbing to release. So they added a frame narrative. It's a little annoying, but not too much so. Of course, all versions diverge radically when it comes to the novel's ending. And critical interpretations also differ as to the movie's sub-textual commentary on the American state of affairs c. 1956.
Is this an allegory about Communism? (Joseph) McCarthyism? Consumerism and mass culture? Good question. As the movie isn't really 'about' any of these things, it supports all the above interpretations and more. Highly recommended.
* Followed by Invasion of the Body Snatchers (1978), Body Snatchers (1993), and The Invasion (2007).