Once, during the Second Temple period, the person holding this position served as deputy to the Nasi, the sage who headed the Supreme Court (Sanhedrin Gedolah). They worked together and were called "Zugot."
The Av Beit Din presided over Supreme Court sessions in the Nasi's absence. Later, the term "Av Beit Din" came to be understood as "Chairman of the Court."
This is how it is still understood today. Each city in Israel typically has a Beit Din, and the city's chief rabbi is the Av Beit Din.