I'm not talking about pure-mathematical functions and such, but rather more philosophical and seemingly less formal ideas, such as consciousness.
The concept of well-definedness is a logico-mathematical one.
A function f(x) is called well-defined if, for all x in the domain of the function, there exists exactly one y in the codomain of the function such that f(x)=y. The definition for multivariable functions is similar. A logical predicate F is a function that maps objects in the domain of discourse to truth values. As such, a predicate is well-defined if, for every object x in the domain of discourse, F(x) is either true or false. (Assuming a two-valued logic-- the many-valued version is likewise similar. The definition generalizes in other ways as well, which I will not mention.)
This plays into philosophical discussions in the following way: For a given natural language predicate like is conscious, we can ask the question of whether this predicate is well-defined with respect to our domain of discourse (e.g. all extant objects). What this question means is, "Does the question 'Is it conscious?' have an answer regardless of what 'it' is?" This is most clearly not the case for so-called vague predicates: for example, there are some people for whom "He is bald" is neither clearly true nor clearly false.
This develops more fully into the philosophy of vagueness and ambiguity. The perceived problem is that we can ask questions which seem like they should have answers, but where there is no answer because the terms in the question are ambiguous or undefined in that particular case.
An object introduced by a definition is called "well-defined" in mathematics if the definition is consistent, i.e. if such an object can exist. The usage is sometimes extended to logic and philosophy. One can put together any conglomerate of properties and start reasoning about objects that satisfy them, but unless there is at least one such object this would be a pointless endeavor. Sometimes it is obvious that a definition covers no objects, as in the case of round squares for example, and sometimes it is a non-trivial question, and one needs to prove that an object is well-defined, for example the limit of a sequence.
In philosophy "existence" is trickier than in mathematics (ideal existence) or in science (physical existence). Correspondingly the issue of well-definiteness is trickier as well. It comes up with some definitions of God for example, which some argue make the concept ill defined because omnipotence is self-contradictory, inconsistent with omnibenevolence given an evil world, omniscience is inconsistent with freedom, etc., see What is the "simple logical truth" that makes omniscience self-contradictory? or Is God either immoral or not omnipotent? Philosophers also argue that some definitions of free will or consciousness make them ill defined or incoherent, see e.g. The incoherence of free will. There are doubts that disembodied souls or philosophical zombies are well defined too (e.g. in the sense of "metaphysical impossibility").
Reasoning about not well-defined objects is something of a wasted effort, unless the point is to demonstrate that they are indeed not well-defined. It gets worse however if in the course of the argument one surreptitiously assumes that the objects of the definition actually exist, that is a logical fallacy of defining into existence. Still making sense of and reasoning about nonexistent objects has a long and colorful history, see Is the use of inconsistent definitions a logical fallacy?
As one comment already explains this isn't a natural part of philosophical vocabulary, it's more naturally a mathematical one.
There, it means given a mathematical object, like a group, or a solution to an equation; or even an operation such as integration or multiplication.
Then they're well-defined if given several different ways to construct the object, or several different ways to do the operation, we end up with the same object.