The breakdown voltage of APDs
The breakdown voltage is the bias voltage at which a single charge carrier (usually an electron) can initiate avalanche breakdown. Avalanche breakdown is when the charge carriers have enough kinetic energy to create additional charge carriers upon impact with the semiconductor lattice. The additional charge carriers are created because the impact gives them enough energy to escape from the lattice.

After the breakdown voltage, the APD is no longer operating in linear mode, i.e., there is no relation like (output current) = (gain)x(photosensitivity)x(input light). Instead, a single charge will result in a rapidly increasing current that is either intentionally quenched or the APD is destroyed. This operation is non-linear: there is a term called saturation that describes the effect of incident photons being ignored while the APD is in avalanche breakdown because the APD current is dominated by the avalanche effect.

A quick internet search will give you more detail than this and have helpful diagrams.