As the title says, what sorts of things would the sailors aboard an ironclad-era warship (specifically mid-1880s if it's too broad otherwise) be expected to do on a day-to-day basis? How would they be organized, in terms of gun crews versus engine crews versus whatever other sorts of specialized crews they might have? I'm interested in pretty much any nation of the time, if there would be differences between them in that respect. Thanks!
The 1880s were an interesting time in ship design, because the traditional divisions of sailors were changing to accommodate the advent of steam propulsion. Most of my comments will apply to the British and American navies, since it's what I'm most interested in.
Navies had since time immemorial been powered by sail, and many early steam vessels also had masts and a full sail rig.
This should not be confused by the mast or masts on a modern ship, which are used mostly for signaling or as sensor platforms. Ironclad masts were used as a primary or secondary power source, as well as a place for sail drill. This was in many ways a logical design evolution, when capital ships were converted to add steam power, though it looks odd to modern eyes.
In any case, the addition of engines meant the addition of engineers, which meant a change in the traditional rank structure of navies and their internal organization. Initially, officers resisted the addition of engines and dirty coal to their ships, even as it was widely recognized that coal and steam power were the future of navies.
(Apologies for the enormous image link, but Turner's "The Fighting Temeraire Tugged to her Last Berth to be Broken Up" is highly symbolic of this understanding.)
Anyhow, the addition of engineers to the traditional structure was a source of tension in many naives. Conservative elements in the British and American navies assigned engineers initially to the ranks of warrant officers, and only reluctantly added them to the ranks of actual commissioned officers. Conversely, up-and-coming officers of a technical bent saw engineering as a growth field, which led to a great deal of tension among younger and older officers.
In terms of internal organization, by the end of the Napoleonic wars the British and American navies had adopted a divisional system, in which an officer was assigned to a set of men on the ship, so that even in a very large ship each man was assured of being known to his own officer. The men were further divided by stations on "watch" by their skills, with the most prestigious jobs for a seaman being work aloft (although there were many other specialized jobs having to do with all stages of running a ship). The change in propulsion attacked directly the boatswain's department, as being in charge of the sails and rigging, and also the self-identity of the Navy. Officers wondered how men would be trained without constant work on the masts and sails, and reactionary elements in the British navy continued to present papers and arguments in favor of sails though the 1890s (sails were officially abolished from new construction in 1887).
In terms of the actual jobs done, sailors tended to do many traditional jobs: scrub decks and fittings, exercise with the guns, hoist in and out boats, re-stock the ship with provisions, take care of compartments, tackle, sails, flags, etc.; but the new jobs done were coaling, maintaining the ship's machinery, and understanding the complicated mechanical changes brought about by boilers and engines and steam power. The men would be organized into divisions under a particular officer, with warrant officers supervising individual work spaces and/or groups of men, who stood regular watches.
Some sources on the transition that are interesting reads:
Sacred Vessels: The Cult of the Battleship and the Rise of the U.S. Navy