  We study standing waves for a nonlinear Schr\"odinger equation on a star
graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an
interaction occurs described by a boundary condition of delta type with
strength $\alpha\leqslant 0$. The nonlinearity is of focusing power type. The
dynamics is given by an equation of the form $ i \frac{d}{dt}\Psi_t = H \Psi_t
- | \Psi_t |^{2\mu} \Psi_t $, where $H$ is the Hamiltonian operator which
generates the linear Schr\"odinger dynamics. We show the existence of several
families of standing waves for every sign of the coupling at the vertex for
every $\omega > \frac{\alpha^2}{N^2}$. Furthermore, we determine the ground
states, as minimizers of the action on the Nehari manifold, and order the
various families. Finally, we show that the ground states are orbitally stable
for every allowed $\omega$ if the nonlinearity is subcritical or critical, and
for $\omega<\omega^\ast$ otherwise.
