These are great questions, because they bring up a pre-modern communications world that is quite foreign to us today. Unfortunately, I don't know of a book that focuses on sailors and communications in the 18th century. There are a number of books on letter writing, and a huge number of books on sailors, but I'm not sure it's a topic that has been the subject of a devoted study by a historian. As a result, I can only answer based on some of the archival research I've done on 18th century merchants.

It seems to me that you're asking two sets of questions—first, specifically about how mail moved between ships at sea (i.e., not when a ship is in port); and second, about how mobile sailors and their correspondents managed to ever stay in touch through letter-writing. I know more about the second set of questions (particularly as it relates to Atlantic commerce and communications), so I'll add some information to help you understand that.

The first thing to notice is that sailors, merchants, and those communicating with them expected that not all their letters would reach one another. This was especially true during wartime, when ships carrying letters were often taken, sunk, or otherwise detained. Sometimes, a ship might simply decide to change course during its voyage, if it heard from passing ships about commercial opportunities elsewhere. One of the strategies that correspondents developed for dealing with this uncertainty (particularly for important letters dealing with business matters) was to send multiple letters by different ships. Additionally, correspondents took careful stock of whether their letters were being received. If you look at almost any 18th century letter, there's a good chance it will begin by noting what they had most recently received from the other person. For example, a typical first line to a letter might read like: "I rec'd yours of the 5th inst." (meaning that the author received his correspondent's letter of the 5th of this month). Another way that correspondents dealt with this was to copy their outgoing mail into letterbooks, so that if they found that the ship they had sent a letter with had not made it to its destination (which they could track by newspapers, word-of-mouth, and yet more letters), then they could resend the message by a different ship. Overall, though, it would be a mistake to assume that 18th-century letter-writing was efficient enough to guarantee delivery.

Second, you ask: "If a person at home wanted to get a letter to a given ship, did they just walk down to the docks and hand it to the first ship headed where they think the target is?" But it's not necessarily the case that correspondents would have a piece of mail waiting to find someone who might be willing to carry it to the recipient. It was very often the reverse. That is, you might hear that a ship is sailing on Wednesday for Port-au-Prince. With this knowledge, you might then take the opportunity to send a letter to your business associate or your family there. That's one reason why Anglophone letters of the time often referred to ships as "opportunities." In this way, letter-writing practices adapted themselves to the realities of commercial traffic.

Third, "If a returning ship is carrying mail from ships encountered at sea and arrives in a port, is there a place for them to drop the letters off for delivery or further carriage to the intended recipient?" Let's say you're in the South Pacific and desperately needed to get in touch with your wife, who lives in Boston. If the next ship to North America was headed to Philadelphia, you could rely on the post office to get the letter to Boston. Many ports in the Anglophone world or British Empire during the 18th century had the equivalent of a post office that would transport letters within empires. But the existence of an intra-imperial postal infrastructure depended on the time and the place. However, if your wife was in a port belonging to the French Empire, such as Cap Francais, while the ship was sailing to British or American Philadelphia, you wouldn't be able to count on the post office. Instead, you might write a letter to your friend in Philadelphia, enclosing a letter to your wife and asking him to forward it to your wife by the next ship sailing to Cap Francais.

Fourth, as to "good will," sending letters on ships was a normal and expected part of life. For commercial voyages, letters were a key way for a ship's captain to communicate with the ship's owners. Ships relied on letter-writing even as letter-writers relied on ships.

Fifth, as for the issue of how mobile sailors and their correspondents located one another, one strategy was to rely on intermediaries. Because sailors moved around more quickly than they could update faraway correspondents of their location, they sometimes relied on a nearby stationary intermediary (who might be able to track the mobile sailor's whereabouts more easily) to forward letters to them. For example, if a ship captain is sailing around the Caribbean and western Atlantic, he might not be able to keep his family in Britain or France apprised of his exact movements, because by the time his letter telling them that he's in Martinique has arrived with his family, he's already moved on to Demerara. However, if the ship captain is also in regular contact with merchant x in Havana, he might ask his family to send him messages "care of merchant x," who would be able to forward them to the next port that the ship captain will be in.

Finally, "it just blows my mind a little seeing notes about French explorers departing Portuguese Brazil and meeting English military vessels in the south Pacific and happening to have aboard letters from England for those specific ships." While this example does seem to have been quite an unusual coincidence, keep in mind that 18th century ships tended to cluster around a limited number of major ports. Even if some ports were supposedly closed to ships of particular nationalities, there were ways of getting around that. So in a sense, even though these ships traversed vast distances, it could seem like a fairly small world when you're moving between the same places repeatedly.

Hope this is helpful.

Great question, I'd like to know the answer as well. Also, while watching Master and Commander it looked like they handed off bags of letters to someone I assume was a merchant? Was this actually done and were they a trusted agent of the Crown?

Is the diary published? What's the title? I have always wondered this too. I'd read stories of people who came to the colonies and would write back about how terrible the conditions were. But I never quite knew how people would get a letter back home. And were stamps used in these cases?

I hope someone has some answers or links to answers!

Much of the British Naval mail exchange systems originated from Falmouth in Cornwall, from the Falmouth Packet Service. Established in 1688 as a hub for spanish mail boats, (Norway 3) Falmouth quickly became an information hub and center for British military orders, frequently sending out mail weekly to Lisbon and San Sebastian, and less frequently to the Mediterranean, Brazil, Surinam, Halifax, and New York; 1200 men were permanently employed by Falmouth for their mailings (Norway 8). In any case, in the 18th century, the British had established a rather functional worldwide mail system.

"History of the Post-Office Packet Service between the years 1793-1815" A.H.Norway (Macmillan, 1895) Found here: https://archive.org/details/cu31924030165538

You, of course, have seen the first example of Clark delivering a letter, on May the 15th, 1787:

abreast of Falmouth and a great number of fishing Boats in Sight —— I wish that on[e] of them would come in board of use that I might send a letter on shore by her to my Betsey but I find it is in Vain to Botany Bay I must goe

Because he's near Falmouth at this time, the logical conclusion is not that he hopes a fishing boat will just pick a letter up and deliver it willy nilly like you describe; he knows that the packet service is very close and will be a definite way to send his letter to Betsey. Unfortunately for clark, they are not stopping in Falmouth, they are going ahead to Botany Bay. But in any case, sailors know that Falmouth is the port where you should send a letter, and they're willing to carry letters for one another.

Two summers ago I visited the Galapagos Islands with my wife and sons. We stopped at Post Office Bay on Floreana Island. There is a barrel on a post there that has been used (at least the site, if not the same barrel) as a way station for mail for quite a long time. We were told that it was started by whalers traveling the Pacific. Anyone traveling past can drop mail in the barrel and is encouraged to pick up and deliver mail near their final destination. It is a tourist thing now, but we picked up and delivered a couple of postcards that were addressed to near our home in Virginia, USA. A bit of living history.

There are some excellent answers here. Many of the ideas expressed were still current when I was at sea some years ago. My family and friends would usually send numbered letters so that I would know if a letter of the sequence had gone astray. Letters would be posted to the office of the shipping company, the shipping company would parcel them up and send them with the company mail to the company's agent in the next expected port of call and the agent would bring them onboard, to be distributed by the Old Man to the crew. This usually happened very quickly after berthing. However, agents in Japan could be very slow at delivering personal mail, considering it unimportant. After a 7 week passage from Brazil to Japan, for example, the ship was visited on arrival by the agents, but the mail wasn't brought onboard until just prior to our sailing five days later. We were not impressed! Mail in the US was also a problem because the Immigration people would refuse to release mail until the arrival process had taken place, which could last for hours as all of the crew had to be lined up and interviewed individually (interviews would not start untiol all of the crew were present and lined up in crew list order..... No matter what time of day it was) which caused quite a lot of bad feeling. Problems could occur if the ship was diverted to another port, and letters could often take weeks or months, if ever, to catch up. Letters home would be given to the Purser who would hand them all to the agent prior to departure. Again, letters would be numbered so that the recipients would know if letters had gone astray.