Some definitions in mathematics are very "explicit" or "constructive". For instance, we can define a palindrome as a word obtained by taking any word W, reversing it to get another word W', and then forming a word WW' or WaW' (where a can represent any symbol in the alphabet being used).
Sometimes, there exists an alternative definition that doesn't suggest directly how you might construct an object satisfying the definition but instead singles out those objects from a much larger universe of possible objects by some characterizing property. For instance, we can consider the set of all possible words and define the involution W -> W' that sends a word to its reversal. Palindromes can then be characterized as those words that are fixed by this map. This latter description doesn't indicate directly how you construct palindromes (although because this example is so simple it is not hard to figure out such a construction!)
Is there an official term for this distinction? It reminds me a little of constructive versus non-constructive proofs -- it would seem more natural to give a non-constructive proof of the existence of an object if the definition of that object is of the latter form. But "constructive definition" seems to have a different meaning related to the constructivist philosophy or movement in mathematics.
Edit: to help clarify the distinction, here's another example. The gamma function can be defined very explicitly, say as
$$ \Gamma(z) = \int_0^{\infty} x^{z-1} e^{-x} \, dx $$
But the Bohr-Mollerup Theorem gives a very "non-constructive" way to define the gamma function (as a function of a positive real variable) -- as the unique such function that is logarithmically convex and satisfies $f(1) = 1$ and $f(x+1) = xf(x)$. From the integral definition, it is easy to compute approximate outputs for non-integer inputs, but it is hard to see how you could from the Bohr-Mollerup version.