I've been reading about the military excursions of the British empire against Napolean and I was wondering how navies in general but the British in particular were able to communicate, coordinate and get logistics in a time without our communication technologies. What kind of technology or cleverness was available at the time?
In the Napoleonic era (and, in fact, the British imperial era more generally), the Royal Navy had bases — the Royal Naval Dockyards — all over the world, which meant they were able to resupply and refit their fleet on-station. A non-comprehensive list, just to illustrate Britain's reach:
That's a pretty broad spread that offers a lot of opportunity for force projection. But your premise is correct: communication was a constant challenge. In general terms, it was facilitated primarily by dispatches sent back to the Admiralty either on navy vessels already bound for Britain, or on chartered packet ships — usually small, lightly armed vessels able to sail faster than most of the Navy's ships.
The best source I can point you to on this question is likely Michael Palmer's Command at Sea: Naval Command and Control Since the Sixteenth Century (2007). But I'd also suggest looking at The Command of the Ocean: A Naval History of Britain, 1649-1815 (2005) by N.A.M. Rodger for a detailed history of the Royal Navy in this period.
In terms of the day-to-day coordination of naval strategy and the actual prosecution of the war, the commanding admirals of each overseas station were given broad orders from the Admiralty and afforded significant latitude in their operations on-station.
Per Palmer, regarding the shift in the late 17th/early 18th century as British naval operations became increasingly globe-spanning:
[Admiral Sir George] Rooke’s instructions took into account not only this global interrelationship but also the unpredictability of campaigns waged in distant seas. Whereas Philip at the time of the Armada and Louis before Barfleur had attempted to exercise centralized control over the operations of their forces, English monarchs did not issue orders to their commanders at sea, but broad instructions meant to guide, rather than to dictate, that left admirals with an enormous amount of both flexibility and responsibility. [p.82]
Similarly relevant in a general sense, on the Portuguese as early (16th century) pioneers of long-distance naval strategy:
There was also a third key to Portuguese success: loyal and able commanders willing to act on their own initiative and monarchs willing to trust those they picked for command. Until the invention of the telegraph in the mid-nineteenth century, states had little choice but to grant commanders operating on distant stations wide powers, often including not only military but also diplomatic and economic responsibilities.
[...]
In the absence of communications, grand strategic command and control worked best if a state’s political and naval leaders shared an understanding of naval capabilities and long-range national goals. Since no state had ever before conducted sustained naval operations on a global scale, the process of developing this understanding was empirical. Throughout the sixteenth century the Portuguese, and other Europeans, gradually grasped the revolutionary change in the nature of sea power and developed concepts to guide the use of naval forces. [p.28]
Hope that helps answer your question. As an aside: in terms of general technological innovation in communication, the French pioneered the optical telegraph — chains of towers transmitting messages by semaphore — during the late 18th and early 19th century, but that's more significant in terms of terrestrial rather than maritime communication.
Mahan, when he was covering the Anglo-Dutch war(about 150 years before Napoleonic war), talks about how the Dutch utilized its merchant ships to warn other merchant ships and colonies know of war (after which the merchant/fishing fleets would rendezvous at a location to be escorted by its naval fleet.
For the most part, prior to wireless communication, Naval officers were reliant on their own intuitions to make judgement and calls. Nelson, for example, set out without knowing where Napoleon and his French fleet were headed. Nelson was essentially relying on his scout ships and news from merchant ships that passed by the French fleet. I would assume most merchant ships of the country would be relied on carrying communcations to other parts of the world, since every so often the fleets need to make port for resupply and the Naval officers would try to make contact with the headquarters in order to give information or try to get information.
The communcations themselves will contain rally points and specific or general orders at which point the Naval Officers must rely on their own cleverness to get whatever done.
/u/k1990 has a really nice answer to this question. One thing that I would also add is that the local commanders were given a wide amount of discretion and asked to use their own judgment in a much broader sense than we think of today. I was recently reading a biography of Admiral Nelson, whose activities in the Mediterranean before and after the Battle of the Nile might help to illustrate this point.
The prelude to the Battle of the Nile was the British fleet getting intelligence that a large fleet with 300+ transports had left Toulon, with the destination unknown. The British had abandoned the Mediterranean at this point, and Nelson was given a squadron and ordered to find the French fleet and destroy it. The clear problem was that he had no idea where to find it, and no clear intelligence about its mission or objectives. There were several spots that could be targets of the French, from Sardinia to Sicily to Malta and points in between, but Nelson had no firm intelligence as to where they might be going. He was sorely lacking in frigates and other small ships for scouting, and had to keep his line-of-battle ships together in case he sighted the French, so he was reliant almost entirely on intelligence from merchant ships to divine French intentions. His fleet was damaged in a storm, during which time Napoleon had descended on Malta and captured it. Nelson then headed to Naples to see if he could gather further intelligence, and was informed by a neutral merchant that Napoleon's fleet had left Malta bound for Alexandria, in Egypt. Unfortunately, that intelligence was faulty and Napoleon had left earlier than the merchant had reported; Nelson immediately ordered the fleet to Alexandria and ignored reports of topsails sighted to the south, which were those of Napoleon's invasion fleet. He reached Alexandria ahead of the French, found it empty, and sailed back to Syracuse in Sicily, where he finally had news of the French invasion of Egypt. He returned there and found and destroyed the French battle fleet in Aboukir Bay, but by that time Napoleon's invasion of France was complete. He was not censured for any of his decisions partly due to the victory of the Nile, but more so because his superiors judged that he had acted with the best possible intelligence he had at the time.
During this time, Nelson sent letters to his commander keeping him apprised of the situation, and also coordinated with the British ambassador to the Kingdom of the Two Sicilied (in Naples) for advice. So the command structure was set up to allow him as the local commander to consult with others, but also gave him wide latitude to perform as he saw fit.