Prompts are the interface for eliciting the capabilities of large language models (LLMs). Understanding their structure and components is critical for analyzing LLM behavior and optimizing performance. However, the field lacks a comprehensive framework for systematic prompt analysis and understanding. We introduce PromptPrism, a linguistically-inspired taxonomy that enables prompt analysis across three hierarchical levels: functional structure, semantic component, and syntactic pattern. We show the practical utility of PromptPrism by applying it to three applications: (1) a taxonomy-guided prompt refinement approach that automatically improves prompt quality and enhances model performance across a range of tasks; (2) a multi-dimensional dataset profiling method that extracts and aggregates structural, semantic, and syntactic characteristics from prompt datasets, enabling comprehensive analysis of prompt distributions and patterns; (3) a controlled experimental framework for prompt sensitivity analysis by quantifying the impact of semantic reordering and delimiter modifications on LLM performance. Our experimental results validate the effectiveness of our taxonomy across these applications, demonstrating that PromptPrism provides a foundation for refining, profiling, and analyzing prompts.