Induced-charge nonlinear electrokinetic phenomena have drawn increasing attention not only due to their fundamental importance but also due to their potential applications for manipulating fluid flows and particles in microfluidics. Such type of nonlinear electrokinetic phenomena is jointly driven by the external electric field and the surface charge induced by the same field on polarizable or conducting surfaces, and is also frequently referred to as induced-charge electrokinetic phenomena. The fluid or particle velocity generated by induced-charge electrokinetics is proportional to the square of the external electric field strength. This is strikingly different from the conventional linear electrokinetics for which the fluid or particle velocity is linearly proportional to the external electric field strength. As a result, the induced-charge electrokinetics can generate larger flow rates and even allows for net flows under AC driving electric fields. Based on the basic theories of electrokinetics and electrostatics, effective electric boundary conditions between liquid-solid interfaces are derived for induced-charge electrokinetics under two situations. These boundary conditions are capable of predicting the induced zeta potentials over surfaces of solids with finite electric properties which are crucial for theoretical characterization of induced-charge electrokinetics. The applications of these two types of boundary conditions are demonstrated by analyzing the DC field driven induced-charge electroosmosis in a slit microchannel embedded with a pair of dielectric blocks and the AC field driven induced-charge electroosmosis around a leaky-dielectric cylinder, respectively. The calculations show that the basic flow patterns for induced-charge electroosmosis are the flow vortices which get stronger as the polarizability and /or the conductivity of solids increase. A complete numerical model is then developed to describe dynamic characteristics of the charging of electric double layer and the associated flows around polarizable dielectrics. The presented model does not invoke various assumptions that can be easily violated in practical applications but usually are made in existing analyses. The comparison with a benchmark solution ensures the validity of the complete model. It is shown that the complete model corroborates the two time scales during the EDL charging revealed in former asymptotic analyses. More importantly, the detailed information inside the EDL during the transient charging is resolved for the first time, which provides insight into the induced-charge electrokinetic phenomena with finite thickness of EDLs.