Diffusion of charged particles in a turbulent, strongly magnetized plasma is considered. The analysis deals with two-dimensional electrostatic fluctuations in the plane perpendicular to an externally imposed homogeneous magnetic field. In the analysis particles are transported in this plane by the gyrocenter drift motion. A nonlinear description is given for absolute diffusion, which yields a Bohm scaling in the case of frozen turbulence or for large amplitudes of electrostatic fluctuations. Particular attention is given to the description of relative diffusion of two charged particles. This process is described by a generalization of nonlinear Brownian motion, including a first stage of very slow initial relative diffusion, followed by a stage of rapid separation, until a final stage is reached where the particles become uncorrelated and classical Brownian-like diffusion is reached asymptotically. The stage of exponential growth (which has been measured in fluid turbulence) corresponds to the 'clump effect' in plasmas: it is a consequence of nonvanishing statistical correlations between particle trajectories. For a drift-wave turbulent spectrum the authors obtained an analytical expression for the Lyapunov exponent for the exponential particle separation, a typical feature of chaotic phenomena. The analysis applies to rather general power-spectra for the turbulent electric field fluctuations.