In a Hebrew leap year, one that carries an intercalated thirteenth month to keep the lunar calendar aligned with the solar seasons, Purim is kept on the fourteenth of the second Adar rather than the first, the date this rule records. The governing principle, stated in the Mishnah, is that the two Adars are treated as adjoining months for this purpose and the later one is given precedence, so the festival is deferred to the month closest to Nisan and Passover rather than split across both.
The first Adar is not left entirely without observance in every custom. Some communities mark a minor, non obligatory Purim Katan on the fourteenth of the first Adar in a leap year, a low key echo of the festival with none of the Megillah reading or other obligations attached to the true date, though this is a custom rather than a universal requirement. The underlying reason for the whole arrangement is the same one that governs the calendar generally, a nineteen year cycle in which seven years carry the extra month, so that Passover, tied by the Torah to the spring barley harvest, does not drift into other seasons the way a purely lunar calendar without intercalation would allow it to. The Islamic calendar, by contrast, carries no such correction at all, which is the specific difference the Quran's own prohibition on intercalation was written against, and it is why an Islamic date like Ramadan cycles slowly through every solar season across a person's lifetime while a Hebrew date like Purim, however it moves between two possible Adars, always lands in roughly the same point of the solar year.