Battle of Trafalgar
Following their earlier vote on 8 October to stay put, some captains were reluctant to leave Cádiz, and as a result they failed to follow Villeneuve's orders closely and the fleet straggled out of the harbour in no particular formation.
It took most of 20 October for Villeneuve to get his fleet organised; it eventually set sail in three columns for the Straits of Gibraltar to the southeast. That same evening, Achille spotted a force of 18 British ships of the line in pursuit. The fleet began to prepare for battle and during the night, they were ordered into a single line. The following day, Nelson's fleet of 27 ships of the line and four frigates was spotted in pursuit from the northwest with the wind behind it. Villeneuve again ordered his fleet into three columns, but soon changed his mind and restored a single line. The result was a sprawling, uneven formation.
At 5:40 a.m. on 21 October, the British were about 21 miles  (34 km) to the northwest of Cape Trafalgar, with the Franco-Spanish fleet between the British and the Cape. About 6 a.m., Nelson gave the order to prepare for battle. At 8 a.m., the British frigate Euryalus, which had been keeping watch on the Combined Fleet overnight, observed the British fleet still "forming the lines" in which it would attack.
At 8 a.m., Villeneuve ordered the fleet to wear together (turn about) and return to Cádiz. This reversed the order of the allied line, placing the rear division under Rear-Admiral Pierre Dumanoir le Pelley in the vanguard. The wind became contrary at this point, often shifting direction. The very light wind rendered manoeuvring virtually impossible for all but the most expert seamen. The inexperienced crews had difficulty with the changing conditions, and it took nearly an hour and a half for Villeneuve's order to be completed. The French and Spanish fleet now formed an uneven, angular crescent, with the slower ships generally to leeward and closer to the shore.
By 11 a.m., Nelson's entire fleet was visible to Villeneuve, drawn up in two parallel columns. The two fleets would be within range of each other within an hour. Villeneuve was concerned at this point about forming up a line, as his ships were unevenly spaced in an irregular formation drawn out nearly 5 miles (8 km) long as Nelson's fleet approached.
As the British drew closer, they could see that the enemy was not sailing in a tight order, but in irregular groups. Nelson could not immediately make out the French flagship as the French and Spanish were not flying command pennants.
Nelson was outnumbered and outgunned, the enemy totalling nearly 30000 men and 2568 guns to Nelson's 17000 men and 2148 guns. The Franco-Spanish fleet also had six more ships of the line, and so could more readily combine their fire. There was no way for some of Nelson's ships to avoid being "doubled on" or even "trebled on".
As the two fleets drew closer, anxiety began to build among officers and sailors; one British sailor described the approach thus:

During this momentous preparation, the human mind had ample time for meditation, for it was evident that the fate of England rested on this battle.

Combat: The battle progressed largely according to Nelson's plan. At 11:45, Nelson sent the flag signal, "England expects that every man will do his duty".

His Lordship came to me on the poop, and after ordering certain signals to be made, about a quarter to noon, he said, "Mr. Pasco, I wish to say to the fleet, 'England confides that every man will do his duty'" and he added "You must be quick, for I have one more to make which is for close action." I replied, "If your Lordship will permit me to substitute 'expects' for 'confides' the signal will soon be completed, because the word 'expects' is in the vocabulary, and 'confides' must be spelt," His Lordship replied, in haste, and with seeming satisfaction, "That will do, Pasco, make it directly."
The term "England" was widely used at the time to refer to the United Kingdom; the British fleet included significant contingents from Ireland, Scotland, and Wales. Unlike the photographic depiction above, this signal would have been shown on the mizzen mast only and would have required 12 lifts.
As the battle opened, the French and Spanish were in a ragged curved line headed north. As planned, the British fleet was approaching the Franco-Spanish line in two columns. Leading the northern, windward column in Victory was Nelson, while Collingwood in the 100-gun Royal Sovereign led the second, leeward, column. The two British columns approached from the west at nearly a right angle to the allied line. Nelson led his column into a feint toward the van of the Franco-Spanish fleet and then abruptly turned toward the actual point of attack. Collingwood altered the course of his column slightly so that the two lines converged at this line of attack.

Just before his column engaged the allied forces, Collingwood said to his officers, "Now, gentlemen, let us do something today which the world may talk of hereafter." Because the winds were very light during the battle, all the ships were moving extremely slowly, and the foremost British ships were under heavy fire from several of the allied ships for almost an hour before their own guns could bear.
At noon, Villeneuve sent the signal "engage the enemy", and Fougueux fired her first trial shot at Royal Sovereign. Royal Sovereign had all sails out and, having recently had her bottom cleaned, outran the rest of the British fleet. As she approached the allied line, she came under fire from Fougueux, Indomptable, San Justo, and San Leandro, before breaking the line just astern of Admiral Alava's flagship Spanish ship Santa Ana, into which she fired a devastating double-shotted raking broadside. On board Victory, Nelson pointed to Royal Sovereign and said, "See how that noble fellow Collingwood carries his ship into action!" At approximately the same moment, Collingwood remarked to his captain, Edward Rotheram, "What would Nelson give to be here?"

The second ship in the British lee column, Belleisle, was engaged by Aigle, Achille, Neptune, and Fougueux; she was soon completely dismasted, unable to manoeuvre and largely unable to fight, as her sails blinded her batteries, but kept flying her flag for 45 minutes until the following British ships came to her rescue.
For 40 minutes, Victory was under fire from Héros, Santísima Trinidad, Redoutable, and Neptune; although many shots went astray, others killed and wounded a number of her crew and shot her wheel away, so that she had to be steered from her tiller belowdecks, all before she could respond. At 12:45, Victory cut the enemy line between Villeneuve's flagship Bucentaure and Redoutable; she came close to Bucentaure with her guns loaded with double or treble shots each, and her 68-pounder carronades loaded with 500 musketballs, she unleashed a devastating treble-shotted raking broadside through Bucentaure's stern which killed and wounded some 200–400 men of the ship's 800-man complement and dismasted the ship. This volley of gunfire from the Victory immediately knocked the French Flagship out of action. Villeneuve thought that boarding would take place, and with the Eagle of his ship in hand, told his men, "I will throw it onto the enemy ship and we will take it back there!" However, Victory engaged the 74-gun Redoutable; Bucentaure was left to the next three ships of the British windward column: Temeraire, Conqueror, and HMS Neptune.

A general mêlée ensued. Victory locked masts with the French Redoutable, whose crew, including a strong infantry corps (with three captains and four lieutenants), gathered for an attempt to board and seize Victory. A musket bullet fired from the mizzentop of Redoutable struck Nelson in the left shoulder, passed through his spine at the sixth and seventh thoracic vertebrae, and lodged two inches below his right scapula in the muscles of his back. Nelson exclaimed, "They finally succeeded, I am dead." He was carried below decks.

Victory's gunners were called on deck to fight boarders, and she ceased firing. The gunners were forced back below decks by French grenades. As the French were preparing to board Victory, Temeraire, the second ship in the British windward column, approached from the starboard bow of Redoutable and fired on the exposed French crew with a carronade, causing many casualties.
At 13:55, the French Captain Lucas of Redoutable, with 99 fit men out of 643 and severely wounded himself, surrendered. The French Bucentaure was isolated by Victory and Temeraire, and then engaged by HMS Neptune, HMS Leviathan, and Conqueror; similarly, Santísima Trinidad was isolated and overwhelmed, surrendering after three hours.