The rapid growth of artificial intelligence (AI) is driving an unprecedented increase in the electricity demand of AI data centers, raising emerging challenges for electric power grids. We provide a comprehensive review and vision of this evolving landscape. Specifically, our paper (i) presents an overview of AI data center infrastructure and its key components, (ii) examines the key characteristics and patterns of electricity demand across the stages of model preparation, training, fine-tuning, and inference, (iii) analyzes the critical challenges that AI data center loads pose to power systems across three interrelated timescales, including long-term planning and interconnection, short-term operation and electricity markets, and real-time dynamics and stability, and (iv) discusses potential solutions from the perspectives of the grid, AI data centers, and AI end-users to address these challenges. By synthesizing current knowledge and outlining future directions, we aim to guide research and development in support of the joint advancement of AI data centers and power systems toward reliable, efficient, and sustainable operation.