Originally, just as with movies, all film for still photography was black-and-white or sepia. It was not until the mid-20th century that color photography became feasible for the masses. Obviously color photography is now open to anyone, and given that we see the world in color, why would we want to kindly remove color from our photographs?
This is not an idle question, because in answering it we identify when to try taking a photograph in black-and-white – or more likely, to try taking the color out of a photograph after the fact using an image-editing program such as Photoshop.
The answer to the above question is that we would want to remove color from a photograph in order to simplify it; to get rid of distracting information so that we can concentrate better on something else. In this it's no different from blurring the background of a portrait or framing a photo in such a way that distracting elements are out of the frame.
So now we know the why, which led us to the when, of black-and-white photography, let's look at the three main situations where we might desaturate (ie remove color from) a photo:
In color, a close-up picture of a person reveals all kinds of distracting details of their face – blemishes, a red nose, uneven skin tones, and so on. Black-and-white photography strips out these details, and this makes it easier for the viewer to concentrate on what the photo reveals about the person's life or personality, which is what portraiture is all about.
In abstract fine art photography
Color is very noticeable in a photo, and it generally trumps things like patterns or shapes. So another use for black-and-white is when a photographer wants to take a photo that is 'about' a pattern or shapes. This type of photograph comes up most often in what we may call 'abstract fine art photography', where there really is not a clear or recognizable subject for the photo.
Note that black-and-white photographs are generally more 'tolerant' of contrast, so fine art photographers will often strip out color from their photographs and then ramp up the contrast (again, using an image editing program).
Whenever the photo is more or less monochromatic anyway
This final category is something of a catchall. While the main use of black-and-white is to remove something distracting from a photo, it's also a good idea to try converting a photo to black-and-white when the photo really has a few bright colors anyway. Another way of looking this is to say that just as color is distracting, a washed-out lack of color is also somewhat distracting.