A.O. Scott wrote the review for the NY Times and I pretty much agree with his assessment that it's almost impossible to summarize this movie. Our main character is Stephane (Gael García Bernal), who has problems differentiating between his dreams and reality. It is also a love story, for he finds a kindred spirit in his neighbor Stephanie (Charlotte Gainsbourg), though naturally there are problems with communication which complicate things. What you have to admire is director Michel Gondry's choice to populate their dreams with handmade objects; he uses papier-mâché, stuffed animals, stop-time animation... it's a refreshing change from movies full of computer animation.
I'll only quote the end of A.O. Scott's review, because until I read it, I couldn't quite articulate my own feeling on it and he seems to put it perfectly:
And so you leave this buoyant, impish movie feeling a little blue: sorry that it had to end and also wishing, perhaps, that it amounted to more. But its fugitive, ephemeral quality is part of its point: dreams, after all, are hard to remember, and perhaps don’t hold the meanings they seem to. Without them, though, our minds would be emptier and our lives much smaller. So while The Science of Sleep may not, in the end, be terribly deep, it is undoubtedly — and deeply — refreshing.