A proof that something exists is "constructive"
if it provides a method for actually constructing it.
Cantor's proof that the real numbers are
uncountable can
be thought of as a *non-constructive* proof that irrational numbers exist. (There are easy constructive proofs, too; but
there are existence theorems with no known constructive
proof).
Obviously, all else being equal, constructive proofs are
better than non-constructive proofs. A few mathematicians
actually reject *all* non-constructive arguments as invalid;
this means, for instance, that the law of the excluded middle (either P or not-P must hold, whatever P is) has to
go; this makes proof by contradiction invalid. See
intuitionistic logic for more information on this.
Most mathematicians are perfectly happy with non-constructive
proofs; however, the constructive approach is popular in
theoretical computer science, both because computer scientists
are less given to abstraction than mathematicians and because
intuitionistic logic turns out to be the right theory for a
theoretical treatment of the foundations of computer science.