With the increasing popularity of online social networks, more and more people are posting information, updating their statuses, and searching for topics there. Massive cross-media big data has been gathered by online social networks, with high dynamics, context-sparsity, and cross-media semantic gaps. In addition, it can be challenging to understand users' search intentions in the setting of social networks. The above problems have brought severe challenges and obstacles to cross-media searches, which also attracted more and more attention on social networks. As a relatively new research topic and interest, the concept, methodology, and overall research idea of cross-media search based on user search intention understanding are not evident in the literature. The research also lacks a unified paradigm and relatively complete research ideas on social networks. To solve these problems, we reviewed more than 100 references based on our preliminary exploration and research experience in this field from the whole process involved. We also detailed methodology, datasets, evaluation indicators, experiment evaluation, and research trends and analyzed the challenges. These works will help beginners quickly establish research ideas and processes in this field, and enable them to focus on algorithm design without paying too much attention to datasets, evaluation metrics, and research frameworks. We believe this review will attract more researchers to focus on social network cross-media search based on user search intention understanding and benefit their work.