Battle for Azeroth is full of novelties, including the sieges. There are two new sieges, one for the Horde and one for the Alliance. The siege of the Alliance is the "Siege of Boralus". Both factions can unlock both sieges, but it's harder for the other.
The siege of Boralus:
The Alliance players must complete the campaigns in the three Alliance areas and a quest row in Boralus. When these are completed, you will receive the achievement "One United Nation" and "The Pride of Kul Tiras". With these achievement, you can start the siege.
For the Horde, it looks a little trickier. You can't even enter the capital of the Tiragardesund Alliance, where the siege takes place. To enter the city, Horde players must play the War Campaign up to the seventh chapter, as it will lead you to Boralus. There, however, you must first reach the Good Will reputation level with the "The Incorporated" faction in order to continue.
The Siege of King Rest:
Horde players must complete each quest in the three Horde Areas and one quest series in Dazaralor. After completing the quests, you will receive two achievements, "The Last Signet" and "Zandalar forever", with which you can now play the siege.