There is a definite difference between two kinds of mathematical proof: construction and existence. A proof is constructive if you claim that there are "things" that satisfy a property and prove this by giving an explicit example of such a "thing." A proof is an existence proof if you claim that there are "things" that satisfy a property but do not show an explicit example of such. The proof of the claim is usually done by showing that if such a thing did not exist, we would have a contradiction. Before the advent of modern computing, many things were shown to exist but not explicitly. One example is from Godel's Incompleteness Theorem, which states that there is a statement in Peano Arithmetic that is true but not provable in Peano Arithmetic (we know its true, but we can't prove it). This was an iconic existence proof, but now there exists an explicit example.