  We study the characteristics of the galaxy cluster samples expected from the
European Space Agency's Euclid satellite and forecast constraints on
cosmological parameters describing a variety of cosmological models. The method
used in this paper, based on the Fisher Matrix approach, is the same one used
to provide the constraints presented in the Euclid Red Book (Laureijs et
al.2011). We describe the analytical approach to compute the selection function
of the photometric and spectroscopic cluster surveys. Based on the photometric
selection function, we forecast the constraints on a number of cosmological
parameter sets corresponding to different extensions of the standard LambdaCDM
model. The dynamical evolution of dark energy will be constrained to Delta
w_0=0.03 and Delta w_a=0.2 with free curvature Omega_k, resulting in a
(w_0,w_a) Figure of Merit (FoM) of 291. Including the Planck CMB covariance
matrix improves the constraints to Delta w_0=0.02, Delta w_a=0.07 and a
FoM=802. The amplitude of primordial non-Gaussianity, parametrised by f_NL,
will be constrained to \Delta f_NL ~ 6.6 for the local shape scenario, from
Euclid clusters alone. Using only Euclid clusters, the growth factor parameter
\gamma, which signals deviations from GR, will be constrained to Delta
\gamma=0.02, and the neutrino density parameter to Delta Omega_\nu=0.0013 (or
Delta \sum m_\nu=0.01). We emphasise that knowledge of the observable--mass
scaling relation will be crucial to constrain cosmological parameters from a
cluster catalogue. The Euclid mission will have a clear advantage in this
respect, thanks to its imaging and spectroscopic capabilities that will enable
internal mass calibration from weak lensing and the dynamics of cluster
galaxies. This information will be further complemented by wide-area
multi-wavelength external cluster surveys that will already be available when
Euclid flies. [Abridged]
