Port Royale 3

To annex a town through peaceful methods you need the following:
-The rank of ‘Navigator’ or higher.
-A reputation with the town’s nation over 90%
-At least 9 businesses in the town
-Currently employing more than 75% of the workers in the town
-Town status of ‘Prosperity’ (75%)
-Popularity of over 90% among the citizens

Once these conditions have been met you need to speak to the viceroy of that nation to take control of the town. Most of these requirements will come naturally as you establish trade routes, carry out tasks and establish workers and businesses. 

This is also required for one of the missions in the trader campaign.

TIP: Reaching the required status of ‘Prosperity’ will most likely be the difficult part of this achievement. To make it easier, set a trade convoy starting on the town you want to take over and set the overall goal of that convoy to prosperity. Also if you have a large supply of goods, wait until you are near the 75% requirement, then dump the goods on the market to give a small boost.