Brain functional imaging data, especially functional magnetic resonance imaging (fMRI) data, have been employed to reflect functional integration of the brain. Alteration in brain functional connectivity (FC) is expected to provide potential biomarkers for classifying or predicting brain disorders. In this paper, we present a comprehensive review in order to provide guidance about the available brain FC measures and typical classification strategies. We survey the state-of-the-art FC analysis methods including widely used static functional connectivity (SFC) and more recently proposed dynamic functional connectivity (DFC). Temporal correlations among regions of interest (ROIs), data-driven spatial network and functional network connectivity (FNC) are often computed to reflect SFC from different angles. SFC can be extended to DFC using a sliding-window framework, and intrinsic connectivity states along the time-varying connectivity patterns are typically extracted using clustering or decomposition approaches. We also briefly summarize window-less DFC approaches. Subsequently, we highlight various strategies for feature selection including the filter, wrapper and embedded methods. In terms of model building, we include traditional classifiers as well as more recently applied deep learning methods. Moreover, we review representative applications with remarkable classification accuracy for psychosis and mood disorders, neurodevelopmental disorder, and neurological disorders using fMRI data. Schizophrenia, bipolar disorder, autism spectrum disorder (ASD), attention deficit hyperactivity disorder (ADHD), Alzheimer's disease and mild cognitive impairment (MCI) are discussed. Finally, challenges in the field are pointed out with respect to the inaccurate diagnosis labeling, the abundant number of possible features and the difficulty in validation. Some suggestions for future work are also provided.