We describe direct electrical measurements of microwave-frequency dynamics in individual nanomagnets that are driven by spin transfer from a DC spin-polarized current. We map out the dynamical stability diagram as a function of current and magnetic field, and we show that spin transfer can produce several different types of magnetic excitations, including small-angle precession, a more complicated large-angle motion, and a high-current state that generates little microwave signal. The large-angle mode can produce a significant emission of microwave energy, as large as 40 times the Johnson-noise background.