Random graph (RG) models play a central role in the complex networks analysis. They help to understand, control, and predict phenomena occurring, for instance, in social networks, biological networks, the Internet, etc. Despite a large number of RG models presented in the literature, there are few concepts underlying them. Instead of trying to classify a wide variety of very dispersed models, we capture and describe concepts they exploit considering preferential attachment, copying principle, hyperbolic geometry, recursively defined structure, edge switching, Monte Carlo sampling, etc. We analyze RG models, extract their basic principles, and build a taxonomy of concepts they are based on. We also discuss how these concepts are combined in RG models and how they work in typical applications like benchmarks, null models, and data anonymization.