Automated program repair (APR) is designed to automate the process of bug-fixing. In recent years, thanks to the rapid development of large language models (LLMs), automated repair has achieved remarkable progress. Advanced APR techniques powered by conversational LLMs, most notably ChatGPT, have exhibited impressive repair abilities and gained increasing popularity due to the capabilities of the underlying LLMs in providing repair feedback and performing iterative patch improvement. Despite the superiority, conversational APR techniques still fail to repair a large number of bugs. For example, a state-of-the-art conversational technique ChatRepair does not correctly repair over half of the single-function bugs in the Defects4J dataset. To understand the effectiveness and failures of conversational LLM-based repair and provide possible directions for improvement, we studied the exemplary ChatRepair with a focus on comparing the effectiveness of its cloze-style and full function repair strategies, assessing its key iterative component for patch improvement, and analyzing the repair failures. Our study has led to a series of findings, which we believe provide key implications for future research.