Context. Our understanding of the formation and evolution of planetary systems is still fragmentary because most of the current data provide limited information about the orbital structure and dynamics of these systems. The knowledge of the orbital properties for a variety of systems and at different ages yields information on planet migration and on star-planet tidal interaction mechanisms. Aims: In this context, a long-term, multi-purpose, observational programme has started with HARPS-N at TNG and aims to characterise the global architectural properties of exoplanetary systems. The goal of this first paper is to fully characterise the orbital properties of the transiting system Qatar-1 as well as the physical properties of the star and the planet. Methods: We exploit HARPS-N high-precision radial velocity measurements obtained during a transit to measure the Rossiter-McLaughlin effect in the Qatar-1 system, and out-of-transit measurements to redetermine the spectroscopic orbit. New photometric-transit light-curves were analysed and a spectroscopic characterisation of the host star atmospheric parameters was performed based on various methods (line equivalent width ratios, spectral synthesis, spectral energy distribution). Results: We achieved a significant improvement in the accuracy of the orbital parameters and derived the spin-orbit alignment of the system; this information, combined with the spectroscopic determination of the host star properties (rotation, Teff, log g, metallicity), allows us to derive the fundamental physical parameters for star and planet (masses and radii). The orbital solution for the Qatar-1 system is consistent with a circular orbit and the system presents a sky-projected obliquity of λ = - 8.4 ± 7.1 deg. The planet, with a mass of 1.33 ± 0.05 MJ, is found to be significantly more massive than previously reported. The host star is confirmed to be metal-rich ([Fe/H] = 0.20 ± 0.10) and slowly rotating (vsinI = 1.7 ± 0.3 km s-1), though moderately active, as indicated by the strong chromospheric emission in the Ca ii H&K line cores (log R'HK ≈ -4.60). Conclusions: We find that the system is well aligned and fits well within the general λ versus Teff trend. We can definitely rule out any significant orbital eccentricity. The evolutionary status of the system is inferred based on gyrochronology, and the present orbital configuration and timescale for orbital decay are discussed in terms of star-planet tidal interactions. Based on observations collected at the Italian Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the frame of the programme Global Architecture of Planetary Systems (GAPS).Also based on observations collected at Asiago Observatory, and Calar Alto Observatory.Full Table 3 is only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/554/A28