The quests are called "tales" in Ghost of Tsushima. Along with Jin Sakai's main tale, there are plenty of side tales related to his allies or locals. There are also the Mythic Tales. These are particularly difficult and rewarding compared to the rest, since they almost always involve a duel. Completing them will always earn you an exceptional piece of equipment or a powerful combat technique.
Completing all Mythic Tales will earn you the Teller of Tales Trophy.
Unlock Mythic Tales
Mythic Tales often involves talking to locals in towns and villages. They'll tell you an incredible story, most of the time linked to a legend, before referring you to a storyteller/musician who can tell you more.
When you talk to the musician, Yamato, a small cutscene will be triggered to accompany his narration. What happens next will depend on the Mythic Tale, but most of the time markers will appear on the map to show you where to go.
We recommend completing all the Mythic Tales, as they are really rewarding.
Be careful, spoilers ahead.