A key challenge for systems neuroscience is the question of how to understand the complex network organization of the brain on the basis of neuroimaging data. Similar challenges exist in other specialist areas of systems biology because complex networks emerging from the interactions between multiple non-trivially interacting agents are found quite ubiquitously in nature, from protein interactomes to ecosystems. We suggest that one way forward for analysis of brain networks will be to quantify aspects of their organization which are likely to be generic properties of a broader class of biological systems. In this introductory review article we will highlight four important aspects of complex systems in general: fractality or scale-invariance; criticality; small-world and related topological attributes; and modularity. For each concept we will provide an accessible introduction, an illustrative data-based example of how it can be used to investigate aspects of brain organization in neuroimaging experiments, and a brief review of how this concept has been applied and developed in other fields of biomedical and physical science. The aim is to provide a didactic, focussed and user-friendly introduction to the concepts of complexity science for neuroscientists and neuroimagers.