The circuit in the diagram is called "clamper" or "DC restoration".
Say the input varies from +5V to -5V. Initially the capacitor has no voltage (Vc = 0). Say that the input input is -5V; as the capacitor has no charge, for a very brief moment the diode gets 5V, so it is forward polarized and there is a big surge of current to the capacitor, which charges until it has 4.4V, so the output is -0.6V. At that point the diode cuts, as it can only conduct while it has 0.6V or 0.7V.
A little time later the input changes to +5V; as the capacitor has 4.4V, the other terminal of the capacitor rises to 9.4V. The diode is reverse polarized, so it doesn't conduct anything. There will be a current to the load, but as long as the load has a large resistance (say 100k or more), the current will be small and the capacitor will keep the 4.4V charge.
Therefore, if the input of the circuit is a square wave of -Vo to +Vo, the output will be a square wave of -0.6V to 2Vo - 0.6V. That is, the circuit will shift the input up, preserving its shape (**PROVIDED** the load is very light), until the lowest voltage of the output is -0.6V.
ps: if the load is significant (say 1kΩ) or the frequency of the input is high, then the output will be distorted and the bias will not be enough to reach DC.