Emphasis on clean energy has led to a widespread focus on lithium-ion batteries. However, a major obstacle is their degradation with several cycles or calendar aging. Battery Management System relies on an essential model-based algorithm to protect the battery from operating outside the safety limit. Thus, this work attempts to answer important research questions on battery models: (1) Are physics-based electrochemical models (EM) robust enough to identify internal cell degradation and abnormal battery behavior? (2) How are the structural simplifications and mathematical order reductions imposed on the EMs and what are their trade-offs? (3) How to apply simplified EM for safer and more efficient battery operation? (4) What are the prospects and obstacles of employing EM-based algorithms in the future? This paper presents a detailed analysis of EM from a modeling and application perspective. The paper introduces battery operating mechanisms, typical failures, and their effects. Followed by an analysis of full order EM (Pseudo 2-Dimensional), and further classification into simpler and advanced reduced-order models. The study demonstrates the gaps in theoretical understanding and their implementation for real-time battery operations such as in thermal management, energy utilization, and fault diagnosis and prognosis.