Mars atmospheric pressure variations induce ground displacements through elastic deformations. The various sensors of the InSight mission were designed in order to be able to understand and correct for these ground deformations induced by atmospheric effects. Particular efforts were made on one hand to avoid direct pressure and wind effects on the seismometer, and on the other hand to have a high performance pressure sensor operating in the same frequency range as the seismometer. As a consequence of these technical achievements and the low background seismic noise of Mars, the InSight mission is opening a new science domain in which the ground displacements can be used to perform atmospheric science. This study presents an analysis of pressure and seismic signals and the relations between them. After a short description of the pressure and seismic sensors, we present an analysis of these signals as a function of local time at the InSight location. Then, the coherent signals recorded by both pressure and seismic sensors are described and interpreted in terms of atmospheric signals and ground deformation processes. Two different methods to remove the pressure effects recorded by SEIS sensors are presented, and their efficiency is estimated and compared. These decorrelation methods allow the pressure generated noise to be reduced by a factor of two during the active day time period. Finally, an analysis of SEIS signals induced by gravity waves demonstrates the interest of ground displacement measurements to characterize their arrival azimuth.