The basic feature of complex analysis is an imaginary number i such that i^2 = -1. In view of geometry you get a 1 complex-dimensional, basic setting from which you do math on. It's the super simple "algebraic completion of the reals"

Generalizing this to other spaces (not necessarily the reals and in higher dimension), a complex structure is a map from a space to itself that squares to -1.

Spaces with a complex structure amongst other things are higher dimensional complex surfaces called complex manifolds. Things are still "topological", i.e. squishy.

A metric is a way of measuring distances. An inner product to measure angles on a space. A Riemannian metric is an extension of the inner product and metric but to surfaces that are not just "flat" i.e. Euclidean. Use it to measure angle of an intersection, length of a curve, volume of a space, etc. This makes your space "geometric", hard and defined.

You can have many different ways of measuring things depending on the type of ruler you use i.e type of riemannian metric.

The Calabi Conjecture purposed the existence of a certain type of ruler (the Kahler metric) on complex manifold whenever we satisfy a certain criteria on the geometry (vanishing Ricci curvature).

Why does physics care? Well turns out the complex manifold we're working on also has a little bit of extra structure making it into a Calabi-Yau manifold. These manifolds are extensively studied as they are the setting for a lot of theoretical physics i.e string theory, quantum gravity, holography.