Why is there a negative piece in the wave for the half-wave rectifier?

In the half-wave rectifier, this corresponds to the slightly-less-cartoonish model of the diode, in which you take into account the $V_{pn}$ voltage drop and the forward and backward resistances of the diode. When the diode is forward-biased at greater than $V_{pn}$ (assume here 0.6 V), the output follows the positive swing of the input. You get a positive hump, but at lower than the signal voltage according to $V_{o} = \left(\frac{R}{R+R_f}\right) (V_{s}-V_{pn})$, taking into account both the voltage drop across the diode, $V_{pn}$, and the voltage division (note that here, because $R_f$ is small, the voltage divider factor is $\sim 1$). Then, during the negative swing of the input, you treat the diode as a large resistance $R_r$, and the output follows the input according to the voltage divider equation, as $V_{o} = \left(\frac{R}{R+Rr}\right)V_s$, where $\frac{R}{R+Rr}$ is now a small number- so you get a much reduced (but non-zero) negative swing on the output.