Graph is a natural data structure for describing complex systems, which contains a set of objects and relationships. Ubiquitous real-life biomedical problems can be modeled as graph analytics tasks. Machine learning, especially deep learning, succeeds in vast bioinformatics scenarios with data represented in Euclidean domain. However, rich relational information between biological elements is retained in the non-Euclidean biomedical graphs, which is not learning friendly to classic machine learning methods. Graph representation learning aims to embed graph into a low-dimensional space while preserving graph topology and node properties. It bridges biomedical graphs and modern machine learning methods and has recently raised widespread interest in both machine learning and bioinformatics communities. In this work, we summarize the advances of graph representation learning and its representative applications in bioinformatics. To provide a comprehensive and structured analysis and perspective, we first categorize and analyze both graph embedding methods (homogeneous graph embedding, heterogeneous graph embedding, attribute graph embedding) and graph neural networks. Furthermore, we summarize their representative applications from molecular level to genomics, pharmaceutical and healthcare systems level. Moreover, we provide open resource platforms and libraries for implementing these graph representation learning methods and discuss the challenges and opportunities of graph representation learning in bioinformatics. This work provides a comprehensive survey of emerging graph representation learning algorithms and their applications in bioinformatics. It is anticipated that it could bring valuable insights for researchers to contribute their knowledge to graph representation learning and future-oriented bioinformatics studies.