We present an overview in the understanding of spin-transfer torque (STT) induced magnetization dynamics in spintorque nano-oscillator (STNO) devices. The STNO contains an in-plane (IP) magnetized free layer and an out-of-plane (OP) magnetized spin polarizing layer. After a brief introduction, we first use mesoscopic micromagnetic simulations, which are based on the Landau-Lifshitz-Gilbert equation including the STT effect, to specify how a spin-torque term may tune the magnetization precession orbits of the free layer, showing that the oscillator frequency is proportional to the current density and the z-component of the free layer magnetization. Next, we propose a pendulum-like model within the macrospin approximation to describe the dynamic properties in such type of STNOs. After that, we further show the procession dynamics of the STNOs excited by IP and OP dual spin-polarizers. Both the numerical simulations and analytical theory indicate that the precession frequency is linearly proportional to the spin-torque of the OP polarizer only and is irrelevant to the spin-torque of the IP polarizer. Finally, a promising approach of coordinate transformation from the laboratory frame to the rotation frame is introduced, by which the nonstationary OP magnetization precession process is therefore transformed into the stationary process in the rotation frame. Through this method, a promising digital frequency shift-key modulation technique is presented, in which the magnetization precession can be well controlled at a given orbit as well as its precession frequency can be tuned with the co-action of spin polarized current and magnetic field (or electric field) pulses.