What is Calabi-Yau (in layman's terms)?

I don't believe a layman's term for them exist :P But the words "Calabi-Yau" refer to the two people who studied them and pioneered in their field. The actuall thing itself is a "Calabi–Yau manifold". They are complex higher dimensional mathematical devices in the study of geometry and are used partly in string theory. That's all I can really tell you, sorry. For an introduction, try searching "Manifold" in Wikipedia.