Rosh Chodesh, the beginning of a new month in the Hebrew Calendar, typically falls on the first day of a month and lasts for a single day.
However, Rosh Chodesh Elul begins on the last day of the preceding month - the 30th day of Av, and lasts for two days.
Shofar blowing during Elul starts on the second day of Rosh Chodesh Elul, corresponding to the first day of Elul.
My previous writings on this subject have been inaccurate.
I thank Cantor Daniel Pincus for bringing this to my attention.