X-ray polarimetry, sometimes alone, and sometimes coupled to spectral and temporal variability measurements and to imaging, allows a wealth of physical phenomena in astrophysics to be studied. X-ray polarimetry investigates the acceleration process, for example, including those typical of magnetic reconnection in solar flares, but also emission in the strong magnetic fields of neutron stars and white dwarfs. It detects scattering in asymmetric structures such as accretion disks and columns, and in the so-called molecular torus and ionization cones. In addition, it allows fundamental physics in regimes of gravity and of magnetic field intensity not accessible to experiments on the Earth to be probed. Finally, models that describe fundamental interactions (e.g. quantum gravity and the extension of the Standard Model) can be tested. We describe in this paper the X-ray Imaging Polarimetry Explorer (XIPE), proposed in June 2012 to the first ESA call for a small mission with a launch in 2017. The proposal was, unfortunately, not selected. To be compliant with this schedule, we designed the payload mostly with existing items. The XIPE proposal takes advantage of the completed phase A of POLARIX for an ASI small mission program that was cancelled, but is different in many aspects: the detectors, the presence of a solar flare polarimeter and photometer and the use of a light platform derived by a mass production for a cluster of satellites. XIPE is composed of two out of the three existing JET-X telescopes with two Gas Pixel Detectors (GPD) filled with a He-DME mixture at their focus. Two additional GPDs filled with a 3-bar Ar-DME mixture always face the Sun to detect polarization from solar flares. The Minimum Detectable Polarization of a 1 mCrab source reaches 14 % in the 2-10 keV band in 10(5) s for pointed observations, and 0.6 % for an X10 class solar flare in the 15-35 keV energy band. The imaging capability is 24 arcsec Half Energy Width (HEW) in a Field of View of 14.7 arcmin x 14.7 arcmin. The spectral resolution is 20 % at 6 keV and the time resolution is 8 mu s. The imaging capabilities of the JET-X optics and of the GPD have been demonstrated by a recent calibration campaign at PANTER X-ray test facility of the Max-Planck-Institut fur extraterrestrische Physik (MPE, Germany). XIPE takes advantage of a low-earth equatorial orbit with Malindi as down-link station and of a Mission Operation Center (MOC) at INPE (Brazil). The data policy is organized with a Core Program that comprises three months of Science Verification Phase and 25 % of net observing time in the following 2 years. A competitive Guest Observer program covers the remaining 75 % of the net observing time.