If you've never heard of it, which I think most of my three readers haven't, it's a movie litmus test that originated here. The test is simple: "Is there at least one conversation between two female characters that isn't about men?"
The test can also be applied to novels, with one big caveat: Novels that use first person or tight third with a male protagonist are less likely to pass, and that's not necessarily something to get excited about. Movies, by contrast, don't generally glue a camera on one person's head and leave it there all film.
One of the better discussions of it is here. I see people object to this test with, "Well, there is this particular novel here, and this is why it doesn't pass the Bechdel's and it's OK." Sure, fine - the canonical example is The Name of the Rose. Setting a novel in a medieval monastery does indeed limit the potential for female-female conversations. But I think this misses the point of the test.* The test is better as a general diagnostic for a body of work - modern movies as a class, or the work of Robert Heinlein, or rom-coms.
The exception I would make to this is in a very particular class of movies/novels. This consists of stories set in the modern world (or a future similar to it) with a wide variety of characters and, if it's a novel, omniscient third narration. I read a novel recently, by an author who I would consider to not have reactionary opinions about the place of women, which had 39 speaking male roles, and only 5 females, the only major character of which is a damsel in distress (who is also the MacGuffin). In the book, it was clear that some women had power, because they are mentioned in passing, yet somehow every major character except the explicitly objectified female character was male. Maybe one or two of these gender decisions were driven by the needs of the story; the rest were like, "Well, I need a random person, and the default personhood is male." Needless to say, it didn't pass Bechdel. It pretty much ruined my enjoyment of a book I would have otherwise liked quite well.
Well, maybe you're saying, so what? Think about (third-person omniscient) books and movies that don't pass the reverse Bechdel's - no male-male conversations. All I can think of is The Women, although there are probably others. However, they are very rare. Start paying attention to Bechdel's and its reverse during your cinematic excursions, and then ask yourself what we can conclude about how genders are portrayed in our society.
*At least, IMHO. You can only put so much in a ten-panel comic strip. So you could reasonably argue for another purpose, and since it's a free country you can use the test however you like. I won't stop you.