From my user profile:

General Histories

• N.A.M. Rodger, The Safeguard of the Sea: A Naval History of Britain, 660-1649: The first volume of Rodger's multi-volume naval history of Britain, this book covers seapower from the earliest days of "England" until the end of the second English Civil War. He includes passages on non-English British navies, though the research in that area is still incomplete and spotty. The series the first comprehensive naval history of England/Britain in nearly a century. Rodger divides his books into four types of chapters: ships; operations; administration; and social history. The books can successfully be read as a narrative straight through, or each chapter can be read sequentially; I have done both. Replete with references and with an excellent bibliography.

• N.A.M. Rodger, The Command of the Ocean: A Naval History of Britain, 1649-1815: The second volume of Rodger's history covers operations, administration, ships, and social history through Napoleon's defeat at Waterloo.

• N.A.M. Rodger, The Wooden World: An Anatomy of the Georgian Navy: An earlier (than the two previous citations) and arguably more accessible introduction to the navy of the mid-18th century, while still providing substantial detail. Establishes Rodger's interest in organizations and organizational history as a way to drive the conversation about navies and their successes or failures.

• Patrick O'Brian, Men-of-War: Life in Nelson's Navy: A slim volume but replete with illustrations, this was intended as a companion to O'Brian's Aubrey-Maturin series, about which more below. Useful to understand details of daily life, ship construction, rigging, etc.

• King, Hattendorf and Estes, A Sea Of Words: A Lexicon and Companion to the Complete Seafaring Tales of Patrick O'Brian: Meant as an atlas and glossary for the O'Brian novels, it's a useful companion for all sorts of naval reading.

• The Social History of English Seamen, 1485-1649, edited by Cheryl A. Fury. A series of essays on the social history of English seamen from the Tudor period onwards. Includes a very interesting chapter on the archaeology of the Mary Rose.

• Royal Tars: The Lower Deck of the Royal Navy, 875-1850 by Brian Lavery. A social history of the lower deck (common crew/sailors) of English and British ships.

• Ian W. Toll, Six Frigates: The Epic History of the Founding of the U.S. Navy: Toll's book is a popular history of the founding of the American navy, but it does spend some time on design and construction and what made the American heavy frigates so successful in limited engagements.

• Tunstall and Tracy, Naval Warfare in the Age of Sail: The Evolution of Fighting Tactics, 1650-1815: Meticulously written and illustrated, this is a deep dive into tactics in British, French, Dutch and Spanish navies. A bit dense for beginners, but rewarding.

• Roy Adkins, Nelson's Trafalgar: The Battle That Changed The World: A recent popular history of Trafalgar, very accessible to novices but with a great attention to detail.

• Adam Nicolson, Sieze the Fire: Heroism, Duty and the Battle of Trafalgar: This is Nicolson's attempt to examine ideals of heroism and the heroic persona set against Trafalgar. It's interesting reading, if not completely successful.

Biographies

• John Sudgen, Nelson: A Dream of Glory and The Sword of Albion: These two books are Sudgen's contribution to the voluminous biographical literature about Horatio Nelson, and well worth a read. A Dream Of Glory in particular takes a very searching look at Nelson's early years, which are often minimized in favor of the more exciting narrative of the Nile/Copenhagen/Trafalgar. Sudgen does become a Nelson fan throughout the books, but his writing is not uncritical and does not tip into hagiography.

• Admiral Lord Thomas Cochrane, The Autobiography of a Seaman: Written in a midcentury style, this covers the life of Thomas Cochrane, 10th Earl of Dundonald, who is often seen as the "real Jack Aubrey." While that comparison is both fair and also lacking in nuance, this autobiography is a good primary source from the horse's (ok, captain's) mouth.

Historial Fiction (that doesn't suck)

• Patrick O'Brian's Aubrey-Maturin series (start with Master and Commander): Both a well-researched story of life aboard British men-of-war and an excellent series of novels in their own right. Later books are written more sloppily and hastily, but you'll want to read them all.

• C.S. Forester's Hornblower series: An early version of the historical naval series. Less well-written than O'Brian's, but with interesting characterization and attention to detail.

There are so many books, you could read till the end of your life. The hardest part: how to arrange reading so that it can be accomplished in a reasonable amount of time.

To do that, I'd recommend:

Read first to get oriented

Mike Rapport's The Napoleonic Wars: A Very Short Introduction (Oxford University Press, 2013)

then you can dig deeper, but it really helps to have that introductory lecture in order to get a global view. Otherwise you'll be knee deep in the details of Marengo and Tilsit with no real sense of how these events are part of some greater whole

1. French Revolution. You won't get where Napoleon or France is coming from without understanding something of the Revolution. It's the soil that grows Napoleon. There are a tremendous number of good general histories, but for "just one" -- I'd pick Simon Schama's Citizens. Schama writes very well and while Marxist historians may quarrel with him for his hostility to the Revolution, he's generally very good.
2. Rise of Napoleon and his regime: Isser Woloch's "Napoleon and his Collaborators". Its striking that for all the books that treat Trafalgar, Austerlitz and Waterloo, most amateurs have very little idea of how Napoleon's regime actually worked. That's important.
3. Napoleon and his wars-- for a one volume treatment, I like Alan Schom's "Napoleon Bonaparte". This is a useful antidote to the fanboy treatment that N often elicits. If you're going to read just one, this biography which is heavily oriented to the military would be my choice. Schom paints what is to my mind a much more accurate picture of Napoleon than those who worship "the great captain". Charles Esdaille's "Napoleon's Wars" is an alternative.
4. Primary sources. There are so many. . . a very good picture of the Napoleonic regime comes from Chancellor Pasquier, but I don't think these memoirs have been translated. For material that has been translated, I'd go with the Memoirs of Caulaincourt; they give a real flavor of what working for Napoleon was like. You can find a portion of these Memoirs published in many sources, including online, as "With Napoleon in Russia". There are many interesting military memoirs that give a flavor for soldiering-- I like Arms and the Woman:The Diaries of Baron Boris Uxkull, 1812-1819 -- but that's just one an enormous genre. Newspapers of the era give a very useful window into what it all felt like at the time. British newspapers are available through a wonderful resource, the British Newspaper Archive

I also very much like a scurrilous bit of gossip, a sort of National Enquirer of the period, from a political writer/propagandist named Goldsmith, who published under the name of "Stewarton". Initially in the pay of the French, he breaks with them and then publishes works like "The Secret History of the Court and Cabinet of St Cloud". Understand that Goldsmith can't be relied on-- he's making some stuff up. But he also was close to some of these personalities, and he heard the gossip, and he reports it. Like the National Enquirer today-- some of this stuff is true. This was a best seller in its time, so if you're trying to understand "what did Britain think of Napoleon, dirt and all" -- this is a good read. Similarly -- there's a very big production of cartoons by such brilliant English political cartoonists as Gillray and Cruikshank. Reading a cartoon and understanding it-- that's hard, and a good test of your calibration to the era.

5. Napoleon's enemies. I can add Roger Knight's Britain Against Napoleon: The Organisation of Victory, 1793–1815 (2013) to u/jschooltiger 's very complete list for Britain.