This project aims to understand Calabi-Yau manifolds by building the necessary mathematics of complex manifolds for defining them. Since Calabi-Yau manifolds can be defined as compact Kähler manifolds with vanishing first Chern class, the majority of this project is centered on establishing what Kähler manifolds and Chern classes are. The treatment of this subject is largely based on chapter 3 of the treatise "Introduction to String Compactification" by A. Font and S. Theisen. Special emphasis is placed on example computations to help illuminate key facets of this topic.