We study the quantum dynamics resulting from preparing a one- dimensional quantum system in the ground state of initially two decoupled parts which are then joined together (local quench). Specifically we focus on the transverse-field Ising chain and compute the time dependence of the magnetization profile, m_l(l) , and correlation functions at the critical point, in the ferromagnetically ordered phase and in the paramagnetic phase. At the critical point we find finite-size scaling forms for the non-equilibrium magnetization and compare predictions of conformal field theory with our numerical results. In the ferromagnetic phase the magnetization profiles are well matched by predictions from a quasi-classical calculation that we describe.