From The Heroes of Might and Magic III wiki

An Adventure map object that forces you to do a quest before being allowed to pass. All your quests will be logged in the Quest Log.

[edit] Tips

If you return after having received your quest but before completing it, you might get a message that contains more hints.

[edit] Possible Quests

• Achieve Experience Level
• Achieve Primary Skills Level (Note: Level can be set independently for each skill)
• Defeat a specific hero
• Defeat a specific monster stack
• Return with Artifacts (Note: They will be lost when completing the quest)
• Return with Resources (Note: They will be lost when completing the quest. Can be a different amount of each resource)
• Be a specific hero
• Be a specific color