World Quests were introduced with the Legion expansion. Each expansion since Legion has added new World Quests that are available only after you earned a certain level or achievement. 

Legion World Quests

Legion World Quests require that you reach level 45 and complete the quest Uniting the Isles. This quest requires that you reach Friendly status with all the factions of the Broken Isles.

Once you unlock Legion World Quests with one character on your Battle.net account, alternate characters can complete Uniting the Isles ignoring the reputation requirement. 

Battle for Azeroth World Quests

Battle for Azeroth World Quests require that you reach level 50, you obtain your Heart of Azeroth, and you complete the quest Uniting Kul Tiras (Alliance) or Uniting Zandalar (Horde) which requires that you reach Friendly status with the factions of Kul Tiras or Zandalar.

Once you unlock Battle for Azeroth World Quests with one character on your Battle.net account, alternate characters can complete Uniting Kul Tiras / Zandalar ignoring the reputation requirement. Alternate characters are not required to obtain the Hearth of Azeroth; however, World Quests that reward Azerite will reward nothing to characters that don't have the Heart of Azeroth.

Shadowlands World Quests

Shadowlands World Quests require that you reach level 60 with one character and you join a Covenant.

Once you unlock Shadowlands World Quests with one character on your Battle.net account, alternate characters can access them as soon as they join a Covenant, which will be possible at level 50 (down from 60). World Quests in Korthia are an exception and still require the alternate character to reach level 60 and unlock Korthia.