How was this dot identified as a mystery planet originally (Pluto in this case)? I was watching a documentary on a mission to study Pluto. At 21:11, an image is presented with an arrow mark pointing a small dot which is now identified as Pluto.
As there are hundreds of dots on the image (I'm not sure what kind of an image this is as I'm not an astrophysicist), how can stars, planets, asteroids and other objects be distinguished in this image? And what kind of an image is this?

 A: One point doesn't tell you much of anything, other than that there is probably something in that direction. Pluto was discovered the same way many asteroids are, by comparing pictures of the same star field at different times, seeing if anything changes.
Below is the discovery image from Wikipedia.

The dot moved across the image, while all the stars stayed the same. Of course this is not conclusive proof. More followup was done to ascertain that we really were seeing something orbiting the Sun far away. After all, it is possible that the two dots are unrelated - they could be fleeting glimpses of two different asteroids, or two unfortunate cosmic rays (though the photographic plates used are less susceptible to this than CCD's).
By the way, they had a special machine back in the old days for doing this sort of search - a blink comparator. Astronomers would mount the (glass) photographic plates in the machine, and rapidly switch the view from one to the other. They simply relied on their eyes to detect differences between the frames.

Once you know the orbit, you can predict precisely where Pluto will be at any given time. This enables you to figure out which "dot" it is in other photographs.
I should also mention that with more information than a single broadband image, one can infer more things about point sources. If you can take images in multiple bands, or better yet do spectroscopy, you can see if the light emitted by the object is consistent with various types of stars, for instance.
