Discipline in the Royal Navy of Nelsons time is often seen as a harsh and unbending code of 'starting', flogging and hanging. But to take punishment out of the context of the times is to miss the comparison between life on land and life at sea during the Georgian period.
The Georgian code of justice was known, with good reason, as the bloody code. On land a man could be given a long jail sentence or transported for life for relatively minor offences , he could be hanged for stealing as little as a handkerchief. Newgate prison routinely kept its prisoners 20 to a cell measuring twenty feet by fifteen.
At sea the rules that the men obeyed were known as the Articles of War. A man could only be hanged for mutiny, treason or desertion. Sodomy was also a capital offence, but few men were prosecuted or hanged for it, and it seems likely that it was a rare occurance on a war ship. The open living space of the men providing few opportunities for privacy.
At sea discipline was relatively easy to maintain. The sailors knew that their lives depended on working together to stop the ship from foundering or being taken by the enemy. This may partly explain why it was possible for a 20 year old to command a ship of experienced seamen, (Nelson was not quite 21 when he was made a post captain, and he was not an exception) as long as the captain didn't endanger their safety the crew were willing to work for him. In port the job was harder, and frequently senior officers would think twice before going below decks.

Flogging itself was carried out by the bosuns mate with a cat of nine tails. The punishment was generally carried out the day after the offence, and the bosuns mate made a new cat for each flogging. The cat itself was heavier than the version used in the army, made of a rope handle about two feet long and an inch in diameter to which the nine tails of line were attached. The line was a quarter inch diameter and about two feet long. The whole thing weighed just under a pound. Once finished it was put into a red baize bag until needed.

The flogging began with the order for all hands to muster aft to witness punishment. The offender was generally lashed to an upturned grating. The officers stood to one side in full dress uniform and the marines lined up aft. The captain would read out the Article of War that the offender had broken and then the order would be given to lay on the dozen lashes. If more than a dozen had been ordered then a second bosuns mate would lay on the next twelve.
The force of the blows can  be shown by the fact that a standard cat of nine tails was easily capable, when wielded by an average man, of braking a one inch by one inch length of knot free pine in half. The effect on the victims back was said to resemble scorched and blackened meat.
The severest form of flogging was a flogging round the fleet. The number of lashes was divided by the number of ships in port and the offender was rowed between ships for each ships company to witness the punishment.
The acceptance of flogging by the sailors to maintain discipline, is hard to measure. In the Great Mutinies at the Nore and Spithead, flogging was not mentioned in the sailors list of complaints. In fact whilst the ships were under control of the mutineers, they ordered floggings to be carried out. But Samuel Leech in his memoirs spends much time railing against the injustice of flogging.