This chapter introduces the fundamental ideas, motivations, and methodology of mathematical modeling as a process for translating real-world systems into quantitative frameworks. It outlines how models serve as simplified representations of complex phenomena, enabling researchers to test hypotheses, explore mechanisms, and make informed predictions. Emphasis is placed on the iterative nature of modeling—from conceptualization and formulation to analysis, calibration, and validation—and on the balance between realism and simplicity that underlies effective model design. Parameter estimation and model analysis are presented as the two core pillars of applied modeling: the former anchors models in data through techniques such as least-squares fitting and information criteria, while the latter reveals system behavior through equilibrium, stability, and bifurcation analysis. The chapter highlights the versatility of ordinary differential equation (ODE) models across disciplines and previews applications in life sciences developed in later chapters, including glucose–insulin regulation in diabetes, tumor–immune interactions in cancer therapy, and multi-species dynamics in the Chesapeake Bay ecosystem. Together, these examples demonstrate how mathematical modeling transforms intuition into quantifiable reasoning, bridging data and theory to advance both scientific understanding and practical decision-making.