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The integration of physical principles into machine learning (ML) frameworks has emerged as a transformative approach in materials science, addressing the limitations of purely data-driven models by incorporating domain knowledge to enhance predictive accuracy, generalizability, and interpretability. This narrative review explores the conceptual taxonomies of physics-integrated ML methods, their applications in materials discovery and design, and the associated challenges in data bias and ethical considerations. Drawing on recent peer-reviewed literature, we classify physics-integration strategies such as physics-informed neural networks (PINNs), hybrid models combining ML with physical simulations, and constraint-based learning, and highlight their roles in solving complex problems such as material property prediction, microstructure analysis, and phase stability. We also examine how data biases in training datasets can propagate errors and inequities in model outputs, and discuss the ethical values underpinning the use of AI in scientific research, including transparency, accountability, and societal impact. The review underscores the potential of these methods to accelerate innovation in materials science while emphasizing the need for rigorous validation and interdisciplinary collaboration. By synthesizing current advancements, this article aims to provide a foundational understanding for researchers and practitioners, paving the way for future developments in this interdisciplinary field.