The main goal of this thesis research is developing a theory to describe the early stages of star formation within magnetized, turbulent molecular clouds, which is a fundamental problem in astrophysics. In giant molecular clouds, supersonic turbulence creates shocks and compresses material to generate overdense structures that can later collapse gravitationally, while the intrinsic magnetic fields in the clouds limit the compression in turbulent shocks and provide support to prestellar cores against self-gravity. Previous numerical simulations had shown promising results that prestellar cores with realistic physical properties can form in shocked regions with the presence of magnetic fields and ambipolar diffusion, but left a big gap in understanding the fundamental mechanism driving prestellar core formation in turbulent, magnetized environments, especially the longstanding puzzle of how these dense, self-gravitating cores form in the diffuse, thermally-supported, and highly magnetized clouds. In this thesis, we firstly adopted both analytic and numerical methods to investigate a one-dimensional C-type shock created by turbulence-accelerated ambipolar diffusion, and we discovered a transient stage that can theoretically generate overdense regions with relatively low magnetic pressure. We then turned to fully three-dimensional MHD simulations with supersonic convergent flows, and quantitatively studied the physical properties of cores formed in the shock-compressed regions, together with the detailed flows leading to core formation. These cores have similar masses and sizes as the observed ones, and form within a timescale comparable to the observed core lifetime. However, we found that am- bipolar diffusion may not be a crucial mechanism for cores to lose magnetic support, because gas in overdense regions preferably flows along the magnetic field lines. We therefore extended the parameter space of our simulations to further examine the anisotropic core formation model. Our results suggest that while prestellar cores are seeded by perturbations from local turbulence, they are built up by collecting surrounding materials anisotropically along the magnetic field lines. To conclude, though turbulence-enhanced ambipolar diffusion can highly reduce the level of magnetization within shock-compressed dense regions, anisotropic contraction may be the key mechanism driving prestellar core formation within turbulent, magnetized giant molecular clouds. This mechanism leads to cores with masses and sizes that are in good agreement with observed prestellar cores.