They are manifolds, with coordinates of each point being n-tuples. Each coordinate in these n-tuple is a complex number, like x+iy. In addition to this they meet two other conditions:
1) They are Kaehler manifolds. As manifolds they have a Riemannian metric, and as complex spaces they have a Hermitian form, and in Kaehler manifolds these two conditions are compatible. I can't get any more specific than that without giving a course in Kaehler manifolds.
2) They satisfy a topological constraint called the vanishing of the first Chern class. This means that they are pretty smooth.
Calabi conjectured that in manifolds like this the Ricci curvature (from Riemannian geometry) would vanish. They would be "locally flat" in a technical sense.
Yau proved Calabi's conjecture and constructed the family of Calabi-Yau manifolds that string theorists use today.
I find Geometry, Topology, and Physics, by M. Nakahara to be an excellent introduction to these topics. It assumes an undergraduate familiarity with set theory, calculus, complex analysis, and linear algebra, but given that it is reasonably self-contained.