Graph signal processing deals with signals which are observed on an irregular graph domain. While many approaches have been developed in classical graph theory to cluster vertices and segment large graphs in a signal independent way, signal localization based approaches to the analysis of data on graph represent a new research direction which is also a key to big data analytics on graphs. To this end, after an overview of the basic definitions in graphs and graph signals, we present and discuss a localized form of the graph Fourier transform. To establish an analogy with classical signal processing, spectral- and vertex-domain definitions of the localization window are given next. The spectral and vertex localization kernels are then related to the wavelet transform, followed by a study of filtering and inversion of the localized graph Fourier transform. For rigour, the analysis of energy representation and frames in the localized graph Fourier transform is extended to the energy forms of vertex-frequency distributions, which operate even without the need to apply localization windows. Another link with classical signal processing is established through the concept of local smoothness, which is subsequently related to the particular paradigm of signal smoothness on graphs. This all represents a comprehensive account of the relation of general vertex-frequency analysis with classical time-frequency analysis, and important but missing link for more advanced applications of graphs signal processing. The theory is supported by illustrative and practically relevant examples.