Agentic AI systems have recently emerged as a critical and transformative approach in artificial intelligence, offering capabilities that extend far beyond traditional AI agents and contemporary generative AI models. This rapid evolution necessitates a clear conceptual and taxonomical understanding to differentiate this new paradigm. Our paper addresses this gap by providing a comprehensive review that establishes a precise definition and taxonomy for "agentic AI," with the aim of distinguishing it from previous AI paradigms. The concepts are gradually introduced, starting with a highlight of its diverse applications across the broader field of engineering. The paper then presents four detailed, state-of-the-art use case applications specifically within electrical engineering. These case studies demonstrate practical impact, ranging from an advanced agentic framework for streamlining complex power system studies and benchmarking to a novel system developed for survival analysis of dynamic pricing strategies in battery swapping stations. Finally, to ensure robust deployment, the paper provides detailed failure mode investigations. From these findings, we derive actionable recommendations for the design and implementation of safe, reliable, and accountable agentic AI systems, offering a critical resource for researchers and practitioners.