During the 18th and 19th century, the British Navy had ships all over the world, and communicating with them had to be difficult. After all, the Battle of New Orleans happened after the British had signed the Treaty of Ghent ending the War of 1812. Although many ship captains or fleet commodores had great freedom to operate tactically within their sphere of influence, what system of communication did the Royal Navy use to pass orders and receive reports from its ships far away? Assuming that orders had to travel by courier on fast sailing ships, did the Navy rely only upon its own ships to pass messages, or did it use fast commercial vessels? Was their a back-up system in case the courier was interecepted? Typically, how long did the process take?
According to Brian Lavery's "Nelson's Navy", communication between the Admiralty and the fleets (at least during the French Revolutionary/Napoleonic Wars) was performed by the navy's own vessels. As noted in a previous answer, these were refered to as despatch vessels or despatch boats, and the role was usually filled by a variety of schooners and cutters attached to the fleets for this purpose. However, as Lavery also notes, any warship that was heading in the right direction might be used. The term "despatch vessel" seems to have referred to a role rather than a type of ship in the same way that, at the time, a "cruiser" was any type of warship sailing on detached operations.
The Admiralty Regulations required that all communication from the commander-in-chief of a fleet or squadron on a foreign station (i.e. outside of home waters) to the Admiralty, was to be sent in duplicate and triplicate "by different conveyances". The aim of which, presumably, was to increase the chances of the communication reaching the intended destination.
The most famous examples of these despatch vessels would be the schooner Pickle and the cutter Entreprenante, which were with the British fleet at Trafalgar. These vessels were both sent home carrying duplicates of Collingwood's despatches following the battle, with the Entreprenante following the Pickle a couple of days later.
In contrast, Nelson sent the first copy of his despatches following the battle of the Nile on board HMS Leander (a 50-gun, forth-rate), as his fleet had lost most of its smaller vessels earlier in the campaign. Another copy was sent on the brig Mutine to Naples. The Leander was captured by a French 74-gun ship and her copy of the despatches was dropped overboard, which illustrates why there was the requirement for multiple copies.
The prior to 1823, packet boats were operated by the post office to move mail, goods and passengers. These operated on a regular schedule from a number of ports around the British coast to locations in Europe, the Americas (including the West Indies) and Africa. Mail to the East Indies would have been carried on HEIC ships. The cross-channel services worked on the most frequent schedule with 2-3 sailings a week, while sailings to more distant locations were once a week or greater. While that sort of frequency would have been acceptable for delivery of personal mail, it would have been unsuitable for the timely delivery of wartime orders and despatches.
After 1823, much of the work of the packet boats was taken over by the Royal Navy (who were trying to find a use for many of the small ships remaining from the Napoleonic period). So the distinction between despatch vessel and packet boat becomes less well defined. Although it's noted that the Navy's existing ships were ill-suited to the essentially civilian role and new ships were built to fit the purpose.
 "Nelson's Navy, The Ships, Men and Organisation, 1793-1815", B. Lavery (Conway, 1989) Pg 263
 "British Warships in the age of sail, 1793-1817", R.Winfield (Seaforth, 2005) Pg 359
 "Regulations and Instructions relating to HM's Service at Sea" (1808) Ch.2 Article XI
 "Nelson's Battles, the Triumph of British Seapower" N.Tracy (Seaforth, 2008) Pg 142
 "History of the Post-Office Packet Service between the years 1793-1815" A.H.Norway (Macmillan, 1895) Pg 8
They sent ships to the various headquarters with messages. Ships would return to their local headquarters to receive orders periodically. Failing that, the HQ would send another ship to the place where a particular ship was operating.
I suppose the navy might have used commercial ships if convenient, but in most cases had to use their own sloops and frigates for the purpose. Commercial ships usually lacked the crews for operating efficiently at high speeds and in foul weather.
As transoceanic trips could take weeks to months, this method took that kind of time.
Before the telegraph, communication was normally by post, which was an office or shack for handling the mail. Military communications were handled right alongside civilian messages and for this reason the post was almost always operated by the government. Each route went over land or by sea as was most convenient and sometimes both. In many cases private contractors would be employed to carry the mail between posts, in which case the packages of mail, called "packets" were sealed in some way to prevent the contractor from reading them. The business of carrying packets was called in England the "packet trade" and was a large business. Special, fast boats called "packet ships" were used to carry the packets and conduct other time-sensitive business.
Naval ships operated according to written orders and would periodically put in to a port to report and collect orders which would be found waiting for them. Military ships used the regular (Royal) post, except in very exceptional circumstances.
In places where maintaining a post was impossible due to cost or political considerations, then the Navy would try to use an embassy as the post. In very remote or dangerous places a standing ship would be used as the post.
Any British ship carrying official mail usually had a special pennant indicating that it was a mail ship.