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.