With the accelerating global trend of population aging, traditional elderly care models are increasingly strained by limited service resources and growing quality demands. As a sustainable and cost-effective alternative, home-based elderly care has garnered widespread attention. However, older adults living at home often face critical challenges, including discontinuous health monitoring and insufficient emotional support. The rapid advancement of Artificial Intelligence (AI) technologies offers new pathways for developing intelligent elderly care solutions. This review systematically examines the major applications of AI in home-based care, focusing on two key domains: intelligent health monitoring and management and smart home environment systems. By analyzing representative research findings and practical implementations both domestically and internationally, this paper identifies core challenges across technical, user-centric, systemic, and ethical dimensions. Furthermore, it outlines future research directions, including personalized modeling, edge intelligence, empathetic human-AI interaction, multimodal data integration, and privacy-preserving strategies. This review aims to provide a theoretical foundation and a comprehensive framework to guide future research and practical advancements in AI-powered smart elderly care.