We present a comprehensive experimental and numerical study of magnetization dynamics in a thin metallic film triggered by single-cycle terahertz pulses of ∼20  MV/m electric field amplitude and ∼1  ps duration. The experimental dynamics is probed using the femtosecond magneto-optical Kerr effect, and it is reproduced numerically using macrospin simulations. The magnetization dynamics can be decomposed in three distinct processes: a coherent precession of the magnetization around the terahertz magnetic field, an ultrafast demagnetization that suddenly changes the anisotropy of the film, and a uniform precession around the equilibrium effective field that is relaxed on the nanosecond time scale, consistent with a Gilbert damping process. Macrospin simulations quantitatively reproduce the observed dynamics, and allow us to predict that novel nonlinear magnetization dynamics regimes can be attained with existing tabletop terahertz sources.