Boy, this thread could turn into a doctoral thesis.
There are a LOT of answers. But to me, the most interesting aspect of surrealism is the blurring of lines between medium, content, and meaning.
For example, this iconinc piece from Marcel Duchamp:
It translates of course to "This is not a pipe," and seems to present a puzzling paradox. It is a pipe, yet it's not, but it is but it isn't. But is it? Isn't it just a
painting of a pipe? Or, getting even further into it, you could look at it and say "This is not a pipe" refers not to the image of the pipe, but to the words themselves, posing no paradox whatsoever.
Similar sorts of mental gymnastics are presented by MC Escher. His exploitation of the limitations of 2-D representations of 3-D structures create seeming paradoxes. Images that do not violate any law of graphic perspective, yet cannot possibly exist. Or can they?
Here's a video demonstrating the reversal of the visual exploitation that Escher relied on.
But that's just one aspect of surrealism. Dali represents a different idea of surrealism. Less about word play and illusion, more about fantasy. Creation of impossible realms with symbolism that challenges the standard definitions of how to use symbolism. But of course they are related in that the visions he creates "can't possibly exist", and yet, there they are, existing in painting form. It's not the same kind of impossibility that Escher presents, but it still begs the essential question of what is impossible if we can imagine it and visualize it.