High-dispersion time-resolved spectroscopy of the unique magnetic cataclysmic variable AE Aqr is presented. A radial velocity analysis of the absorption lines yields K2=168.7+/-1km/s . Substantial deviations of the radial velocity curve from a sinusoid are interpreted in terms of intensity variations over the secondary star's surface. A complex rotational velocity curve as a function of orbital phase is detected which has a modulation frequency of twice the orbital frequency, leading to an estimate of the binary inclination angle that is close to 70{deg}. The minimum and maximum rotational velocities are used to indirectly derive a mass ratio of q=0.6 and a radial velocity semi-amplitude of the white dwarf of K1=101+/-3km/s. We present an atmospheric temperature indicator, based on the absorption-line ratio of FeI and CrI lines, whose variation indicates that the secondary star varies from K0 to K4 as a function of orbital phase. The ephemeris of the system has been revised, using more than 1000 radial velocity measurements, published over nearly five decades. From the derived radial velocity semi-amplitudes and the estimated inclination angle, we calculate that the masses of the stars are M1=0.63+/-0.05M_{sun}; M2=0.37+/-0.04M{sun} , and their separation is a=2.33+/-0.02R{sun}. Our analysis indicates the presence of a late-type star whose radius is larger, by a factor of nearly 2, than the radius of a normal main-sequence star of the same mass. Finally, we discuss the possibility that the measured variations in the rotational velocity, temperature and spectral type of the secondary star as functions of orbital phase may, like the radial velocity variations, be attributable to regions of enhanced absorption on the star's surface.