Ingrid Bergman's character marries a local of Stromboli, without really knowing what it would be like when she gets there, which in the 1950s was quite a remote location! The film is all about how she tries to adapt to life on the island. Needless to say she has a miserable time throughout the whole of the film, and cries a lot. The one fantastic thing about this film is seeing what life was like on the island during the time (assuming that Rossellini portrayed it somewhat realistically), with the major employment the fishing industry, and in particular what a somewhat major eruption of the volcano may be like to experience. During all this she discovers that her husband is abusive, and when she gets pregnant tries to escape the island with the help of another good looking fisherman. Not entirely sure what happens at the end of the film, she tries to escape by going to Ginostra (via the summit) and getting a boat, but ends up collapsed on the summit of the volcano crying (again). In the end we don't know whether she commits suicide, stays on the island, or manages to leave. That pretty much sums up the whole film.
On the version of the film that I bought there are several extras, including an overview of the film at the time and controversy surrounding it, involving the relationship between Ingrid Bergman and Roberto Rossillini, but also of a competing film called Volcano (set on Vulcano) which was also in production. This extra was probably better than the film itself!
My recommendation: if you have been to Stromboli and are interested in how it may have looked 60 odd years ago, give it a go, because it certainly is fascinating. I certainly wouldn't consider this a movie classic however.