In the original Battlestar Glalactica Starbuck and Apollo were the character 's names, in the reboot they were callsigns.
"Kara Thrace" was her name, "Starbuck" was her callsign, it is the same character , at least until the end of season 3, not precisely sure what she was after that.
Basiclly pilots have a callsign that is quicker and easier to say than the pilots full name, and also reduces confusion if the squadron has people with simular sounding names.
Similar to how "Lee Adama" was known as "Apollo" in the same show or "Steve Rogers" is known as "Captain America" in Marvel products and "Bruce Wayne" is "Batman" in DC.
This is common practice amoungst fighter squadrons and a lot of other people\groups\services adopted simular practices, for example the current President of the USA has the callsign "Air Force One" if he is on board any fixed wing aircraft that aircraft can identify itself as "Air Force One" and anyone will know the President is on board.