  (abridged) We investigate the regularity of cluster pressure profiles with
REXCESS, a representative sample of 33 local clusters observed with XMM-Newton.
The sample spans a mass range of 10^14 M_sun <M_500<10^15 M_sun. We derive an
average profile from observations scaled by mass and z according to the
standard self-similar model, and find that the dispersion about the mean is
remarkably low beyond 0.2R_500, but increases towards the centre. Deviations
about the mean are related to both the mass and the thermo-dynamical state of
the cluster. Unrelaxed systems have systematically shallower profiles while
cooling core systems are more concentrated. The scaled profiles exhibit a
residual mass dependence with a slope of about 0.12; however, the departure
from standard scaling decreases with radius and is consistent with zero at
R_500. The scatter in the core and departure from self-similar mass scaling is
smaller compared to that of the entropy profiles, showing that the pressure is
the quantity least affected by dynamical history and non-gravitational physics.
Comparison with several state of the art numerical simulations shows good
agreement outside the core. Combining the observational data below R_500 with
simulation data above, we derive the universal pressure profile, that, in an
analytical form, defines the physical pressure profile of clusters as a
function of mass and z up to the cluster 'boundary'. Using this profile and the
observed pressure profiles, we investigate the scaling properties of the
integrated Compton parameter Y, considering both the spherically integrated
quantity and the cylindrically integrated quantity, directly related to the
Sunyaev-Zel'dovich (SZ) effect signal. We further derive the expected
Y_SZ-M_500 and Y_SZ-L_X relations for any aperture.
