  We explore the potential of the $e^+ e^-$ colliders to discover dark matter
and determine its properties such as mass and the spin. For this purpose we
study spin zero and spin one-half cases of dark matter, $D$ which belongs to
$SU(2)$ weak doublet and therefore has the charged doublet partner, $D^+$. For
the case of scalar dark matter we chose Inert Doublet Model, while for the case
of fermion dark matter we suggest the new minimal fermion dark matter model
with only three parameters. We choose two benchmarks for the models under study
which provide the correct amount of observed DM relic density and consistent
with the current DM searches. We focus on the particular process $e^+ e^- \to
D^+ D^- \to D D W^+ W^- \to DD(q \bar{q})(\mu^\pm\nu)$ at 500 GeV ILC collider
which gives rise to the "di-jet +$\mu$ + missing $E_T$" signature and study it
at the level of fast detector simulation, taking into account Bremsstrahlung
and ISR effects. We have found that two kinematical observables -- the energy
of the muon, $E_\mu$, and the angular distribution of $W$-boson, reconstructed
from di-jet, $\cos\theta_{jj}$ are very powerful in determination of DM mass
and spin, respectively. In particular we have demonstrated that in case of
fermion DM, the masses can be measured with a few percent accuracy already at
500 fb$^{-1}$ integrated luminosity. At the same time, the scalar DM model
which has about an order of magnitude lower signal, requires about factor of 40
higher luminosity to reach the same accuracy in the mass measurement. We have
found that one can distinguish fermion and scalar DM scenarios with about 2
ab$^{-1}$ total integrated luminosity or less without using the information on
the cross sections for benchmarks under study.
