There are 4 types of Quests in Genshin Impact, Archon Quests, Story Quests, World Quests, and Commissions. See list of quests here.

• Archon Quests tell the main story of the Traveler and Paimon as they travel the world of Teyvat in search of the Traveler's missing twin sibling.
• Story Quests consists of chapters related to the background and personalities of each playable character.
• Most World Quests are unlocked by speaking to NPCs through Teyvat. They give insight into the various locations and stories of the world.
• Commissions are usually simple daily quests players can complete for rewards, but some have a unique story tied to them.