A trading post (abbreviated TP) is an inexpensive form of colonization that connects a province to the trading system without settling it.
Effects of Trading Posts
The primary effect of trading posts is their creation of trade income. They create trade taxes for their owning country, and send generated trade to their center of trade, but otherwise yield no income. The amount of both kinds of trade is dependent on the level of the trading post.
Trading posts assert an ownership claim on their province. Other countries cannot colonize the province, and countries without access to the owning country cannot move military units into it. The owning country also gains any adjacent waters as national waters.
Trading posts increase the supply in their province.
Trading posts have a population of 10 men per level. Their population is exceptional in that it neither grows nor shrinks via normal population growth. The only effect of the population is when a province with a trading post is colonized: the men in the trading post are added to the 100 colonists in the level 1 colony when it is created. This gives the colony a small head start in population.
In places with a bad climate (tropical Central America and all of Africa), it is often a good idea to send a trading post to a province first, even though you intend to colonize immediately. These places have negative growth rates for colonies. Sending a trader first gives the colony an extra 10 men, cheaply. Thus, even with negative growth, the colony will require only 10 colonists, instead of 11, to become a city.
Burning Trading Posts
Any time a country controls an enemy trading post, it has the option of burning it. For the human player to do so, select the province screen: the "Burn Trading Post" button is at the bottom. When you burn a trading post, you lose the 1% warscore it was worth. The province becomes unowned, and anyone may colonize it.
AI countries always burn trading posts the day after they capture them. (There is always time for one try at peace.) Rebels never burn them. Natives often destroy trading posts when they rise, but if they don't, they do not burn them. Low aggression natives will usually just knock down the trading post a level or two.