Aims: Until recently, most high-resolution spectroscopic studies of the Galactic thin and thick 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 the thin and thick discs, using the first internal data release of the Gaia-ESO survey (GES iDR1). Methods: We used the spectra of around 2000 FGK dwarfs and giants from the GES iDR1, obtained at resolutions of up to R ~ 20 000 with the FLAMES/GIRAFFE spectrograph. We derive and discuss the abundances of eight elements (Mg, Al, Si, Ca, Ti, Fe, Cr, Ni, and Y). Results: We show that the trends of these elemental abundances with iron are very similar to those in the solar neighbourhood. We find a natural division between α-rich and α-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. This separation is visible for most α-elements and for aluminium. 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 the first paper from this series for our sample, 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 (4 to 12 kpc) and vertical (0 to 3.5 kpc) gradients in metallicity, iron, four α-element abundances, and aluminium for the two populations, taking into account the identified correlation between RGC and | Z |. Similarly to other works, a radial metallicity gradient is found in the thin disc. The positive radial individual [α/M] gradients found are at variance from the gradients observed in the RAVE survey. The thin disc also hosts a negative vertical metallicity gradient in the solar cylinder, accompanied by positive individual [α/M] and [Al/M] gradients. The thick-disc, on the other hand, presents no radial metallicity gradient, a shallower vertical metallicity gradient than the thin-disc, an α-elements-to-iron radial gradient in the opposite sense than that of the thin disc, and positive vertical individual [α/M] and [Al/M] gradients. We examine several thick-disc formation scenarii in the light of these radial and vertical trends. Based on observations collected at ESO telescopes under Gaia-ESO survey programme.Full Table 2 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/572/A33