  We present results from thirteen cosmological simulations that explore the
parameter space of the "Evolution and Assembly of GaLaxies and their
Environments" (EAGLE) simulation project. Four of the simulations follow the
evolution of a periodic cube L = 50 cMpc on a side, and each employs a
different subgrid model of the energetic feedback associated with star
formation. The relevant parameters were adjusted so that the simulations each
reproduce the observed galaxy stellar mass function at z = 0.1. Three of the
simulations fail to form disc galaxies as extended as observed, and we show
analytically that this is a consequence of numerical radiative losses that
reduce the efficiency of stellar feedback in high-density gas. Such losses are
greatly reduced in the fourth simulation - the EAGLE reference model - by
injecting more energy in higher density gas. This model produces galaxies with
the observed size distribution, and also reproduces many galaxy scaling
relations. In the remaining nine simulations, a single parameter or process of
the reference model was varied at a time. We find that the properties of
galaxies with stellar mass <~ M* (the "knee" of the galaxy stellar mass
function) are largely governed by feedback associated with star formation,
while those of more massive galaxies are also controlled by feedback from
accretion onto their central black holes. Both processes must be efficient in
order to reproduce the observed galaxy population. In general, simulations that
have been calibrated to reproduce the low-redshift galaxy stellar mass function
will still not form realistic galaxies, but the additional requirement that
galaxy sizes be acceptable leads to agreement with a large range of
observables.
