Mathematics operates in terms of propositions. A mathematical definition is a proposition in which the mathematical concept being defined occurs. The concept is well-defined if the definition allows it to be identified uniquely.

Examples:

q is the natural number divisible by 5.
With this definition, q isn't well-defined: there is more than one such number.
Q is the set of natural numbers divisible by 5
Q is well-defined: a set is defined by its members and although this set is infinite we know exactly what its members are
Q is the set of natural numbers that occur in the decimal expansion of pi
this is a less well-defined set::we know it is unique, and we have a procedure to find all of its members, but is it possible to find out whether or not any given number is in it? I don't actually know
Q is the set of natural numbers that do not occur in the decimal expansion of pi
here, we don't even have a procedure to list the members of the set; for all we know, it may be empty
T is the set of Turing machines that halt on the empty input
this set is undecidable: there exists no method to determine for any given Turing machine whether it is in this set
and so forth. Different schools of mathematics put different limits on what they still consider acceptable definitions - for instance, intuitionism is rather stricter than standard mathematics.