Most high-resolution spectroscopic studies of the Galactic discs were mostly confined to objects in the solar vicinity. Here we aim at enlarging the volume in which individual chemical abundances are used to characterise both discs, using the first internal data release of the Gaia-ESO survey. We derive and discuss the abundances of eight elements (Mg, Al, Si, Ca, Ti, Fe, Cr, Ni, and Y). The trends of these elemental abundances with iron are very similar to those in the solar neighbourhood. We find a natural division between alpha-rich and alpha-poor stars, best seen in the bimodality of the [Mg/M] distributions in bins of metallicity, which we attribute to thick- and thin-disc sequences, respectively. With the possible exception of Al, the observed dispersion around the trends is well described by the expected errors, leaving little room for astrophysical dispersion. Using previously derived distances from Recio-Blanco et al. (2014b), we further find that the thick-disc is more extended vertically and is more centrally concentrated towards the inner Galaxy than the thin-disc, which indicates a shorter scale-length. We derive the radial and vertical gradients in metallicity, iron, four alpha-element abundances, and Al for the two populations, taking into account the identified correlation between R_GC and |Z|. Radial metallicity gradient is found in the thin disc. The positive radial individual [alpha/M] gradients found are at variance from the gradients observed in the RAVE survey. The thin disc also hosts a negative vertical metallicity gradient, accompanied by positive individual [alpha/M] and [Al/M] gradients. The thick-disc, presents no radial metallicity gradient, a shallower vertical metallicity gradient than the thin-disc, an alpha-elements-to-iron radial gradient in the opposite sense than that of the thin disc, and positive vertical individual [alpha/M] and [Al/M] gradients. ; Comment: 24 pages, 10 figures