A complex manifold of (complex) dimension n is a topological space which looks locally, around each point, like an open subset of \C^n, and on which it makes sense to talk about holomorphic (or analytic) functions. Setting n=1 we get complex curves (also called Riemann surfaces, as they have 2 real dimensions).
A complex curve in a complex manifold X can be described in two different ways: as a holomorphic map from the curve to X, or as the zero locus of a bunch of holomorphic functions on X. Or, in physics-speak, it can be described as the worldsheet of a string or as a D-brane.
These different points of view suggest different ways to "count" such complex curves in a fixed manifold X; the first leading to "Gromov-Witten invariants" and the second to the more recent invariants of Maulik-Nekrasov-Okounkov-Pandharipande (MNOP) when X has dimension 3 (the dimension relevant to string theory). The MNOP conjecture is an extraordinary and mysterious conjecture relating these two "counts".
I will try to explain all this, and then a third method of counting curves, also partly motivated by string theory. If time allows I will also explain how this sheds light on a fourth, conjectural count of curves, due to Gopakumar and Vafa, that would be the "best" solution to this problem, in some sense.
This is joint work with Rahul Pandharipande.