Well-defined normally means what you defined does not lead to any contradictions.
For example, I can try to define a function f from real numbers to real numbers s.t.
f(0) = 1
f(x + y) = f(x) + f(y)
but this is not well defined because
f(0) = 1 and
f(0) = f(0 + 0) = f(0) + f(0) = 1 + 1 = 2
Leading to a contradiction.
For functions proving it is well-defined means proving that each input maps to exactly one output.
This is just a small example. Normally well-defined has other meanings depending on context.