In this talk we will introduce persistent homology, a fundamental tool in Topological Data Analysis (TDA), which captures and analyzes the topological structure of data across multiple scales. We will build mathematical intuition for how persistent homology tracks features such as connected components and loops in data, and how these features persist or vanish as the scale changes. A key focus will be on cycle matching, which allows us to compare topological features across different spaces or datasets, shedding light on the consistency of structures in data transformations. Throughout the talk, we will relate these concepts to practical applications, namely showing how TDA provides insights into complex datasets by revealing underlying topological patterns. The goal is to give a foundational understanding of the basic tools of TDA and their relevance in modern data analysis.