On the island of Delos, Leto had twins, Apollo and Artemis.
But usually, twins have a time in between before they are born. Not sure if this is the same in mythology.
So which was older?
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Most accounts that make the distinction agree that it was Artemis who was born first.
From Pseudo-Apollodorus, Bibliotheca 1. 21:
She [Leto] finally reached Delos and gave birth to Artemis, who thereupon helped her deliver Apollon. Artemis became a practised huntress and remained a virgin.
From Callimachus, Hymn 3 to Artemis 22:
Even in the hour when I [Artemis] was born the Moirai (Fates) ordained that I should be their helper [women in childbirth], forasmuch as my mother suffered no pain either when she gave me birth or when she carried me win her womb, but without travail put me from her body.
Both quotes from Theoi.
Note that some accounts imply that there was a separation between the birth of the twins (see e.g. the references in Woman's Power, Man's Game: Essays on Classical Antiquity in Honor of Joy K. King), but this is not explicitly stated anywhere, as far as I know.