So this will be brief as I only have about 45 minutes before class, but we have time enough to write an answer and beat the Spaniards prep for teaching too.
I think that the most important reason for Nelson's success at the Nile was that he had the entire summer to talk with his captains as he chased Napoleon back and forth across the Med. He spent a great deal of time with each of his captains, in groups and individually, and often their lieutenants or other officers, in sharing breakfasts, dinners and in informal conversation about how he would beat the French fleet once they caught it. He spent enough time with his captains that they knew how he would approach nearly any tactical situation, and just as importantly, he made sure his captains knew that he would support them in taking their own initiative.
This answer that I previously wrote goes more into depth about the evolution of Nelson's tactical thinking over time: https://www.reddit.com/r/AskHistorians/comments/3ebdk5/how_did_nelsons_tactics_work_at_trafalgar/
To quote:
Nelson had studied [breaking the line], and saw the potential for [its} decisive use. In the battle of Cape St. Vincent (1797), when Nelson was still a captain, he broke his ship out of the line of battle without orders so he could engage the Spanish van (front part of their fleet), engaging three Spanish ships with his one and taking two of them as prizes. (The exact number of ships that came to his aid is in dispute, but his 74-gun HMS Captain engaged ships of 130, 112 and 80 guns for a period of time.) Nelson could have been censured for breaking the line without orders, and could quite possibly have lost his ship in the process. The British admiral, Sir John Jervis (later created Earl St. Vincent) did not reprimand Nelson, but also did not mention his action in dispatches. (Nelson himself used his tactics for propaganda purposes, but I'm getting away from the point.)
Nelson also used the tactic of concentrating the strength of his fleet upon a smaller portion of the enemy's line in his tactics at the Battle of the Nile (sometimes called the Battle of Aboukir Bay) in 1798. That battle came after a long and frustrating summer of chasing the French from one end of the Mediterranean to another, which provided Nelson (now an admiral) with the time necessary to meet with his captains and make his tactical intentions known. When Nelson finally caught up with the French fleet, it was at anchor, but he proceeded to attack immediately with the intention of pitting his ships 2-1 or 3-1 against the front of the French line. On his own initiative, Thomas Foley, captain of HMS Goliath, noticed that there was room between the French ships and the shoal water to the west, and passed down the west side of the French line. Other ships followed, so the French line was essentially doubled, allowing the British to anchor, beat ships into submission, weigh anchor and proceed down the line.
The key to the battle of the Nile was that Nelson knew he was likely to be outnumbered, but he specifically planned to attack by concentrating the whole weight of his fleet against a small portion of the French line, defeating it in detail, then moving on to another portion of the line. Nelson knew that the navy was very much the junior service in France, and that individual captains were not in a situation where they could take much initiative to respond to their admiral's orders. He also intended to engage the French admiral directly with his own ship, but in the event of the actual battle L'Orient was about at the point in the line where the British fleet ended (with doubling the French fleet).
As to the other part of your question:
what skills did he possess that lead us to characterise him as a brilliant admiral?
Nelson was not a great seaman (he was often seasick, and had questionable skills as a sailor). Nor was he modest or self-effacing; on the contrary, he actively worked to cultivate a cult of personality around himself. He could be quite trivial and often rude to people he thought didn't matter.
On the other hand, he was a superb leader of men and a decent judge of character; he seems to have made it a point to get to know his captains not only to impress upon them his goals and desires but also to take the measure of the men and to know whom he could trust and not trust. He managed some difficult campaigns with skill (particularly the siege and eventual conquest of Malta, with the aid of allies) and had an uncanny skill at what today we'd call "situational awareness," which he displayed both at St. Vincent and at Copenhagen.
He also seems to have had that spark of charisma that is hard to define no matter how many leadership seminars one attends; he could be utterly charming and laser-focused on a person in a moment, and seemed to inspire intense personal loyalty. This is an account of Arthur Wellesley meeting Nelson before Trafalgar:
"Why," said the Duke, "I am not surprised at such instances, for Lord Nelson was, in different circumstances, two quite different men, as I myself can vouch, though I only saw him once in my life, and for, perhaps, an hour.
"It was soon after I returned from India. I went to the Colonial Office in Downing Street, and there I was shown into a little waiting-room on the right hand, where I found, also waiting to see the Secretary of State, a gentleman whom, from his likeness to his pictures and the loss of an arm, I immediately recognized as Lord Nelson.
"He could not know who I was, but he entered at once into conversation with me, if I can call it conversation, for it was almost all on his side, and all about himself, and in really a style so vain and so silly as to surprise and almost disgust me. I suppose something that I happened to say may have made him guess that I was somebody, and he went out of the room for a moment, I have no doubt to ask the office-keeper who I was, for when he came back he was altogether a different man, both in manner and matter. All that I had thought a charlatan style had vanished, and he talked of the state of this country and of the aspect and probabilities of affiars on the Continent with a good sense, and a knowledge of subjects both at home and abroad that surprised me equally and more agreeably than the first part of our interview had done; in fact, he talked like an officer and a stateman.
"The Secretary of State kept us long waiting, and certainly for the last half or three quarters of an hour I don't know that I ever had a conversation that interested me more. Now, if the Secretary of State had been punctual, and admitted Lord Nelson in the first quarter of an hour, I should have had the same impression of a light and trivial character that other people have had, but luckily I saw enough to be satisfied that he was really a very superior man; but certainly a more sudden and complete metamorphosis I never saw."
Your twofold question has a few answers. By most accounts (in particular, that of a man who was on HMS Majestic who wrote to his mother in Whitehaven after the battle detailing what happened), the French fleet was unprepared. Nelson was considerably more aggressive and had already spent several months discussing possible battle plans against the Napoleonic fleet with his captains. When Nelson's fleet found Brueys, the French fleet was essentially 'napping' and a considerable number of sailors were ashore digging for fresh water as well as setting up batteries. While it doesn't provide much of an excuse for how poorly the French fleet performed, the result is that by the time the French could respond Nelson had already sprung into action. A part of this was Brueys's belief that the English fleet wouldn't attack late during the day (Nelson's fleet found the French sometime between 5 and 6 PM).
The Majestic sailor's account highlights one of the most well known traits of Nelson's command - when Majestic's captain died during the battle, the lieutenant took command without so much as a whisper of a delay. Nelson had always encouraged amongst his captains, and thus the lieutenants under those captains, a tendency for independent action and in-the-moment decision making; it's one of the reasons Nelson was so famously loved by those he commanded (see: Dudley Pope's "Life in Nelson's Navy"). Pope summarised this commandment of Nelson's by writing that Nelson's captains would not have questions asked of them if they were to succeed although 'no answer [would] be sufficient' in the case of failure. Pope also argued that Nelson's success at the Nile was less the actual battle than it was Nelson's willingness to chase the French fleet in spite of the fact that he was risking professional ruin.
Nelson is considered so brilliant today for a combination of all these reasons as well as a healthy disrespect of the rules. One might recall that as a commodore, Nelson had disobeyed all the rules at Cape St Vincent - something he instilled in his men. His rapport with his men, which was certainly not hindered by his willingness to enter battle himself just like the common man, was legendary as well.
On this last point you are free to disagree with me, but Nelson's death at Trafalgar and ultimately his martyrdom, is likely the biggest catalyst Nelson ever had in immortalising him as one of the greatest commanders the Royal Navy has ever had.
Some sources on the matter in case you're curious:
Does anybody have any good suggestions for further reading on Nelson, naval tactics of the time, and his twist on/departure from those tactics?