Boyd’s Bible Dictionary:
(for a consul). A Roman official, beneath a consul, who exercised authority in a province. Appointed by the senate (Acts 13:7; 19:38).
Concise Bible Dictionary:
One who acts as a consul in a province. The word ἀνθύπατος, translated “deputy” in the AV, shows the accuracy of Luke in giving this title to the governor of places to whom it belonged (Acts 13:7-8,12; Acts 18:12; Acts 19:38).