Why does the half moon appear to have a "nose"? Yesterday evening, the moon from my place looked like this:



Basically, instead of the crescent shape I am used to it looked like there was three tips: one at each extremity (as usual) and one in the middle between the two first ones. This picture shows the rough shape of the visible moon, with arrows pointing at all three tips:



What is the explanation of this?
Here's an image I captured from a Moon globe app on my iPad.



From your description, I think you may be referring to the southern rim of Mare Tranquillitatis, the Sea of Tranquility, perhaps extending a little into the Rimae Hypatia. This is an area which is lower than the surrounding mountains and craters, into which the shadow is encroaching.

Incidentally, Mare Tranquillitatis is the region where Neil Armstrong and Buzz Aldrin landed on the Apollo 11 mission!