Proteins, as essential biomolecules, play a central role in biological processes, including metabolic reactions and DNA replication. Accurate prediction of their properties and functions is crucial in biological applications. Recent development of protein language models (pLMs) with supervised fine tuning provides a promising solution to this problem. However, the fine-tuned model is tailored for particular downstream prediction task, and achieving general-purpose protein understanding remains a challenge. In this paper, we introduce Structure-Enhanced Protein Instruction Tuning (SEPIT) framework to bridge this gap. Our approach incorporates a novel structure-aware module into pLMs to enrich their structural knowledge, and subsequently integrates these enhanced pLMs with large language models (LLMs) to advance protein understanding. In this framework, we propose a novel instruction tuning pipeline. First, we warm up the enhanced pLMs using contrastive learning and structure denoising. Then, caption-based instructions are used to establish a basic understanding of proteins. Finally, we refine this understanding by employing a mixture of experts (MoEs) to capture more complex properties and functional information with the same number of activated parameters. Moreover, we construct the largest and most comprehensive protein instruction dataset to date, which allows us to train and evaluate the general-purpose protein understanding model. Extensive experiments on both open-ended generation and closed-set answer tasks demonstrate the superior performance of SEPIT over both closed-source general LLMs and open-source LLMs trained with protein knowledge.