What does "unique modulo" mean? "The square root of a square number modulo 257 is unique modulo 257"
(This is something that I have to prove or invalidate)
I don't understand what 'is unique modulo 257' means?

It means that if $x^2 \equiv z \pmod {257}$ and $y^2 \equiv z \pmod {257}$ then $x \equiv y \pmod {257}$.
Generally "a thing with property X is unique modulo ..." means that if two things have property X (in this case, their square is equivalent to the same number mod 257) then they are equivalent in whatever the equivalence relation following the word "modulo" is.
