The bird's beak is formed in the LOCOS (LOCal Oxidation of Silicon) Process when oxidizing some regions more than others. The former regions are called the field oxide and the later less oxidized are the transistors' gate oxide. The bird's beak region is the one where the oxide thickness gradually shrinks from large (field oxide) to small (gate oxide) (because of its shape's similarity with a bird's beak).
I don't know exactly what is the "weak" bird's beak region but i assume that it is the thinner side of the beak, namely that close to the gate oxide