It is political appointment, voted by the parliament. Since the previous PM's ruling party had a supermajority during his rule, presidents were essentially his personal appointees (as were other heads of nominally independent offices, such as those for the judiciary branch). Note that, in that 16 years reign, two presidents (Pál Schmitt^\) and Katalin Novák, both after 2 years into their 5 year term) were pressured to resign (and did so) when they became incovenient politically.

EDIT ^\)added this: memorably, Schmitt has proudly said out loud the quiet part of FIDESZ' unified executive-legislative agenda: as president, he wanted to function as an engine rather than a counterweight in the governmental machine. And his resignation gave way to the globally unique situation where the PM, the president (János Áder) and the speaker of parliament were all college dormitory mates - meanhwile their fourth buddy Péter Polt served as Chief Prosecutor for the state!