The etrog has two features- the top of the etrog, known as the pitom (which the gemara calls the shoshanta), and the bottom of the etrog, the oketz.
If the etrog grew with the pitom, the pitom is considered to be part of the etrog, and therefore if it falls off, it may disqualify the etrog, as the etrog would be considered to be "haser" (missing something). It is important to emphasize that we are referring to the green stem which is attached to the small, wood-like tip. If only the wooden tip falls off, the etrog is not considered to be "haser," and is still valid. There are some etrogim which grow without a pitom. In this case, it is perfectly valid.
The Shulhan Aruch (438:8) writes that if the entire oketz is missing, and one only sees the hole, then the etrog is pasul (disqualified). This only applies on the first day of the Hag; on the other days, the etrog is valid. However, Hachan Ovadia writes that if one has the oketz, he should preferably glue it or attach it which a needle to the etrog.
Summary: If the green or yellow stem of the etrog, known as the pitom, breaks, the etrog is invalid. An etrog which grew without a pitom is valid. If the oketz, i.e., the bottom, wooden stem of the etrog is missing, then the etrog is invalid for the first day.
