We have obtained deep far- (FUV) and near-ultraviolet (NUV) images of the inner region of the dense globular cluster M15 with the Advanced Camera for Surveys on board the Hubble Space Telescope. The FUV-NUV colour-magnitude diagram shows a well defined track of horizontal branch stars, as well as a trail of blue stragglers and white dwarfs. The main sequence turn-off is clearly visible at FUV~23.5 mag and FUV-NUV~3 mag, and the main sequence stars form a prominent track that extends at least two magnitudes below the main sequence turn-off. As such, this is the deepest FUV-NUV colour-magnitude diagram of a globular cluster presented so far. Cataclysmic variable and blue straggler candidates are the most centrally concentrated stellar populations, which might either be an effect of mass segregation or reflect the preferred birthplace in the dense cluster core of such dynamically-formed objects. We find 41 FUV sources that exhibit significant variability. We classify the variables based on an analysis of their UV colours and variability properties. We find four previously known RR Lyrae and 13 further RR Lyrae candidates, one known Cepheid and six further candidates, six cataclysmic variables, one known and one probable SX Phoenicis star, and the well known low-mass X-ray binary AC211. Our analysis represents the first detection of SX Phoenicis pulsations in the FUV. We find that Cepheids, RR Lyraes and SX Phoenicis exhibit massive variability amplitudes in this waveband (several mags). Comment: 47 pages, including 14 figures, accepted for publication in ApJ, Table 7 corrected, final version including ApJ corrections