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These arrows work much like Light Arrows in The Wind Waker, destroying most normal enemies in a single hit. They will deal significant damage against Guardians. One arrow is given to Link by Beedle on his Customer Appreciation Day. It's unclear if that's first time the player talks to Beedle on a particular day of the month, or after some amount of interactions with Beedle or some combination thereof.
Ancient arrows make all normal enemies vanish instantly, not even leaving weapons behind. To bosses and sub-bosses, the arrow deals 50 damage, plus the bow attack power. To guardians, the arrow depletes 1/3 of their health.
The best place to obtain these is either opening chests around Hyrule Castle, or just buying them from the Akkala Ancient Tech Lab.
The Ancient Arrow is the result of a regular Arrow being combined with an Ancient dagger, which is a secret bit of Ancient Technology discovered by Robbie. The first was created solely by him, 20-30 years before Jerrin joined him from Kakariko Village.
Hyrule Compendium entry of the Ancient Arrow.