You need to own land to progress with the main quest. This guide on How To Own A Settlement In Mount & Blade II Bannerlord will tell you what you need to do in order to own a settlement so you can complete the objective for either an Imperial or Non-Imperial Settlement for the Dragon Banner quest.
There is a more passive route than the one I used personally. You can join another Kingdom and work alongside their forces, completing quests, aiding in battle. Eventually they may offer you a town or a castle as payment for your assistance. This is probably an easier route than the aggressive technique outlined below, but it will take some time.
How To Own A Settlement In Mount & Blade II BannerlordFor me personally, this is the best method. Fighting. A settlement, one that you must capture for the quest, is specifically talking about castles. These aren’t the big cities with taverns and such, the smaller castles where you can visit nobles. You must lay siege to the castle and defeat it.
To lay siege to a castle you must first be at war with the faction. Use a hostile action at a nearby village to do this quick enough. Then you can lay siege to the castle. During the siege you can build siege weaponry to destroy the walls, giving you a faster way in. Be careful, the full Garrison will come out and attack you eventually, usually supported by a nearby allied army. Look for a castle with a small garrison to make this easier.
Once the castle falls and it comes under your control, that will complete the objective for owning either an imperial or non-imperial settlement.