Our by-laws specify that a VP must take it (it's part of the job of VP, taking over when the president isn't there). The VP probably has been at the board meetings and knows what's going on, whereas a new nominee would have a lot of catch-up.
Doesn't it usually go to the VP? At least in our organization, that's how it works. If the President can't finish the year (which hasn't happened), the VP takes over. In our organization, the VP (co-VPs) are the ones who will assume the presidency the following year anyway, so it's a natural succession.
If the VP can't do it, then nominations are accepted and the voting process is invoked for both a new president and VP.
We've never lost a president (though many have thought about it. ;-0) but we have had VPs drop out--in fact, three in the last five years.
In general, the President tenders her resignation in writing or verbally to the board (effectively creating a motion that needs to be addressed). The board then votes on the motion. Assuming that the motion passes, the president is released from her obligations and the position of president is then vacant. Usually, the VP steps into the president's position for the remainder of the term.
If the VP is unable to unwilling to do so, your board follows the by-laws to fill the empty seat for the remainder of the term.
What is the "official" protocol for a President to resign her position? She accepted a job just before school started and she works in another school district. Her son will be moving to her "workplace" so she officially can't serve on the PTO at this point, since she will have no children at the school.
