Question

does the square root of 256 represent the positive square root of 256, the negative square root of 256, or both?

Answer 1

Both, because a negative squared and a positive squared would both still give you a positive product. For example: -4 x -4 = 16 and 4 x 4 = 16 too. 

Answer 2

As long as you're in the realm of mathematics, it represents both.

If it arises as the solution to a real-world situation, then one of the
solutions may have no practical application, and you'll discard it
because it's not a helpful one.