The output characteristics of a continuous-wave diode pumped free-running double-clad ytterbium-doped fiber laser are investigated. In the experiment, not only self-pulsing is observed, but also self-mode-locking is observed for the first time in the free-running fiber laser. The physical mechanism responsible for these phenomena is analyzed. The analysis reveals that the signal is absorbed in the weakly pumped or non-pumped part of the active fiber, which leads the self-pulsing to be present, and self-mode-locking is induced by axial mode beating and self-phase modulation, which are enhanced by nonlinear effects, stimulated Raman scattering, and stimulated Brillouin scattering. The pulse width of self-pulsing decreases as pump power increases. The pulse width of self-mode-locking pulses becomes narrow with the increase of the pump power. When the pump power is higher than 8 W, self-pulsing and self-mode-locking disappear. The interval between adjacent self-mode-locking pulses is 224 ns, corresponding to the cavity round-trip time. The maximum and the minimum durations (full width at half maximum) of self-mode locked pulses are about 35.0 ns and 6.3 ns respectively, and the maximum and minimum spectral widths (full width at half maximum) of signal light are about 7.05 nm and 2.01 nm respectively.