The Mandelbrot set can be defined as a limit, and in that sense you can be absolutely certain whether a given point belongs to it or not. For example, the point c=0 belongs to the Mandelbrot set. An interesting remark in Wikipedia:
Originally Posted by Cougar
The intersection of M with the real axis is precisely the interval [-2, 0.25]. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family [...] In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set.
That's in theory. But for most points in the complex plane it's very difficult to tell in practice whether they belong to the set or not. The best you can do is get an approximate answer by having a computer do a couple of iterations, and classifing the points based on how close or far the associated sequence is to converging after those iterations (this is often coded with colours).
In short: pictures like the one Frog March posted are only approximations to the actual set.