I apologize if this comment is not considered in-depth enough for the subreddit, but I'll give it my best shot.

First, we know that the Chinese were using some sort of postal system well before the 1800s, if only because Marco Polo (13th century) wrote about it. Marco Polo describes in his Travels that the postal system comprised of a series of waypoints that the letters would be delivered to-- a messenger would travel to the designated postal area, hand off the letter, and then another messenger with a fresh horse would then deliver it to the next area, and so on until it finally reached its destination.

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At each station four hundred good horses are kept in constant readiness, in order that all messengers going and coming upon the business of the grand khan, and all ambassadors, may have relays, and, leaving their jaded horses, be supplied with fresh ones. Even in mountainous districts, remote from the great roads, where there were no villages, and the towns are far distant from each other, his majesty has equally caused buildings of the same kind to be erected, furnished with every thing necessary, and provided with the usual establishment of horses. He sends people to dwell upon the spot, in order to cultivate the land, and attend to the service of the post; by which means large villages are formed. In consequence of these regulations, ambassadors to the court, and the royal messengers, go and return through every province and kingdom of the empire with the greatest convenience and facility...

[...]

In the intermediate space between the post-houses, there are small villages settled at the distance of every three miles, which may contain, one with another, about forty cottages. In these are stationed the foot-messengers, likewise employed in the service of his majesty. They were girdles around their waists, to which several small bells are attached, in order that their coming may be perceived at a distance; and as they run only three miles, that is, from one of these footstations to another next adjoining, the noise serves to give notice of their approach, and preparation is accordingly made by a fresh courier to proceed with the packet instantly upon the arrival of the former... at each of these three-mile stations there is a clerk, whose business it is to note the day and hour at wich one courier arrives and the other departs; which is likewise done at all the post-houses.

- The Travels of Marco Polo, the Venetian (translated by Thomas Wright, 2002)

While Marco Polo is obviously describing a messenger relay system being used by the Great Khan to correspond with other important peoples, that might not be considered sufficient enough of an answer, because it seems you're asking how normal people would write letters to one another.

Your average layperson was not able to use this postal system. It is only being used to deliver correspondence to government officials or would-be government officials. In that case, it's just a matter of knowing who the recipient is and which district they reside in.

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There was a well-established old order which met the needs of the Chinese state and people on a pre-modern basis. The state had its long established official post, a far flung network of post stations on major routes over which horse relays carried couriers with official communications, as well as personages, and goods.

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Although the system was very efficient by pre-modern standards, it had become increasingly inadequate by the late Ch'ing period. Besides, the official postal system served exclusively the needs of the government; postal service for private individuals varied with the locale. A city-dweller could enjoy the luxury of having his mail delivered or picked up by a courier at his house or his place of business and transmitted safely to any part of China at reasonable cost by the commercial letter agencies, (min) hsin-chii, called hongs by Westerners. For those living in the interior or in certain rural areas, however, courier facilities were either infrequent or non-existent. Very often, letters were sent by special messengers or were entrusted to muleteers, chair-bearers, travelers and the like because couriers and letter agencies tended to operate only in large towns and along routes where there was a sufficient volume of commercial correspondence to make it profitable.

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Because of the importance attached to official correspondence, no Chinese government extended the use of the postal system to the people. The I-chan, not unlike the Pony Express in America, was reserved for important affairs of state...

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In line with tradition, the Ch'ing Official Post served the triple function of postal transmission, transportation, and travel service. Main post routes radiated from Peking, connecting it with all provincial capitals where resided the governors-general, governors, commissioners, and commanders-in chief. Subsidiary post routes formed numerous networks around each district. Along the five main routes, at intervals of about 100 li, were post stations with horses and other facilities, and on subsidiary routes, at intervals of approximately fifteen to twenty-five li, were foot-courier stations.

- Postal Communication in China and Its Modernization 1860-1896by Ying-Wan Cheng; East Asian Research Center Harvard University (1970)

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So, if you're a low-ranking Chinese official who lives in a huge metropolitan area, and you want to deliver your letter to another official, what you do is go to the postal station and hand the courier your letter. You tell him who it is to be delivered to and which district they live in. He then runs off to the next station, and hands it off to the next guy. So on and so forth until it eventually reaches the correct station. Once there, a foot courier then goes to the official's house or place of work and hands the letter to them.