In mathematics, a constructive proof of existence leaves you with a method to actually find an instance of the object whose existance is being asserted. By insisting on constructive proof, existence becomes a stronger notion.

A constructive definition is a definition of (a) mathematical object(s) that can be constructively proved to exist.

At least, this summarizes my understanding of an article on L. E. J. Brouwer in a weekly magazine 20 years ago.

One of the methods of proof that Intuisionists/Constructivists reject is that of the excluded middle. This law implies that everything is either true or false. Typical examples of this can be found in propositional logic. However this excludes the possibility of "maybe" or "I don't know" or "uncertain" or "undetermined".

Many of Brouwer's work on constructivism come across as fairly mystical writings. One of the few "assumptions" that he allows is the existence of the natural numbers, which he believes we all have some underlying conciousness of.

An interesting result of having to be constructive in proofs is that all functions end up being continuous. (Something many students I know would be very happy about;) Other important consequences include not being able to use the Axiom of Choice and hence Zorn's Lemma, which is generally used to give the natural topology on the reals. This also disallows proofs of the existence of non-principal ultrafilters.

Con*struct"ive (?), a. [Cf. F. constructif.]

1.

Having ability to construct or form; employed in construction; as, to exhibit constructive power.

The constructive fingers of Watts. Emerson.

2.

Derived from, or depending on, construction or interpretation; not directly expressed, but inferred.

Constructive crimes Law, acts having effects analogous to those of some statutory or common law crimes; as, constructive treason. Constructive crimes are no longer recognized by the courts. -- Constructive notice, notice imputed by construction of law. -- Constructive trust, a trust which may be assumed to exist, though no actual mention of it be made.