As the most significant naval battle of WWI, Jutland had a great influence on naval design in the following years. Jutland showed navies that future battles would be fought at long range, that flash was hugely dangerous, that underwater protection was vital, and, somewhat dubiously, that ships needed heavy armour.
One of the most important lessons of Jutland was that battles would be fought at longer ranges than previously expected. During the run-up to WWI, the Royal Navy was planning to fight engagements at ranges of 8,000 yards or below. At Jutland, thanks to the threat from torpedoes and the German unwillingness to close, ranges were closer to 15-20,000 yards. This had several effects on design philosophies. Firstly, there was a general need to improve fire control to cope with such ranges. This included adding larger rangefinders, as well as the use of better fire-control computers. However, this was much less important than the effect on armour design. Pre-war expectations were that, at close ranges, a ship could chose to fire high-explosive shells to ruin the unarmoured parts of its target, or fire armour piercing shells to damage the protected citadel. As such, pre-war designs had a medium-armoured citadel, and thin armour at the bow and stern. However, with these longer ranges, this was no longer feasible, as the accuracy needed to do this was unobtainable. Instead, commanders were forced to fire armour piercing shells at long-ranges. This resulted in the all-or-nothing armour scheme (which the USN had been working on at the time of Jutland thanks to their planning for long-range engagements in the Pacific). In this, ships had a thick armoured citadel, and unarmoured ends, with sufficient protected buoyancy to keep them afloat should the ends flood. The move to longer ranges also resulted in a move towards heavier guns, capable of doing more at these longer ranges. Such guns were also to have lower muzzle velocities with heavy shells, as this increased accuracy at long range.
Flash was shown to be a problem by the explosion of the three British battlecruisers, Queen Mary, Indefatigable and Invincible. On these ships, hits to the turret started fires in the cordite propellant. These propagated down the hoists to the magazines, igniting these and destroying the ships. To avoid this, several steps were taken. Flash-tight doors were tested thoroughly, with greatly improved designs put into place. Screens were installed between guns in turrets (called 'fearnought' screens in the RN), preventing fires from spreading throughout the turret. Some other navies, notably the French, would replace these screens with bulkheads, making the protection much more substantial. Working and handling chambers were installed on all turrets, and on casemated secondary armaments. Cables, especially those using lead, were removed from magazines, as dripping lead from burning cables was found to ignite charges. Crews were issued with anti-flash equipment, to prevent burns. Pre-war designs had the magazines above the shell rooms, as it was thought that the shells, filled with the unstable picric acid, were the greatest risk. However, the experience of Jutland showed that it was actually the propellant charges that were the big danger. As a result, post-Jutland designs moved the shell rooms above the magazines. HMS Hood was, unfortunately, too far along in construction for this change to be made to her, and this may well have contributed to her loss in an explosion in 1941.
The explosions of the three battlecruisers was claimed by senior figures in the RN to be the result of the battlecruisers being underarmoured. This is provably false, as no shells penetrated either the magazines or machinery rooms of any other ship in the battle - the closest was a hit to Barham which sent shell splinters into her 6in magazine. The claim that the ships were underarmoured, made by Beatty and Jellicoe, instead served to cover up their negligence with respect to magazine procedures. However, it would be an influential one. British battleship designs in the years following Jutland tended to emphasise armour over other concerns, a pattern generally replicated in the designs of other navies. That said, the view of the RN's Director of Naval Construction, Tennyson d'Eyncourt was that light armour had proved to protect ships well. It kept the explosions from shells outboard, and prevented most splinter damage. As such, there were still lightly armoured designs for capital ships in the interwar period, most notably the planned American Lexington class. Jutland also proved the need for fast fleet scouts, leading to the rise of the heavy cruiser following the Washington treaty, which was similarly lightly armoured.
Finally, underwater protection was shown to be necessary. Torpedo hits to the British Marlborough and German Seydlitz, while survivable, effectively took both ships out of the fight. Underwater damage from shells, especially to the German Lutzow, proved to be heavily damaging. This was combined with losses to mines and torpedoes outside of the battle, including the losses of Audacious and Inflexible. These experiences showed that flooding was the most important threat to capital ships, and that protecting against it was key. Ships built and designed post-Jutland received considerably greater underwater protection than their predecessors. Great attention was also paid to reducing flooding paths, by increasing subdivision and reducing penetrations of bulkheads for pipes and the like.
Sources:
David K Brown, The Grand Fleet: Warship Design and Development 1906-1922, Seaforth, 2010
David K Brown, Nelson to Vanguard: Warship Design and Development 1923-1945, Seaforth, 2010
Norman Friedman, Fighting the Great War at Sea: Strategy, Tactics and Technology, Seaforth, 2014
Broadly speaking, battleship design is about balancing three things: Armor, Armament, and Speed. Heavy, thick armor came at a cost to speed; high speed meant a cost to armour protection. Large guns gave an advantage to range and hitting power, but at a cost to armor and speed, and the temptation to use them in the line of battle.
British battleships tended to emphasize speed and armament at the expense of armor; the newest ships in the Grand Fleet were the Queen Elizabeth-class battleships, carrying 8 15" main guns and could make 25 knots, but with an armor belt of only 13". This culminated in the designs of the battlecruisers, ships with battleship armaments (13.5" main guns on the Lion-Class) but with cruiser armor and speed. German warships, on the other hand, emphasized armor and survivability at the expense of speed and armament. Compare the QE class to the Bayern's speed of 21 knots but with almost double the deck plating... A serious flaw in the Royal Navy's design.
Interestingly, these choice end up "balancing" each other's fleets into the stalemate that ultimately defines Jutland. The battlecruiser commander (VAdm David Beatty) gives in to temptation and puts his ships into battleline against the more heavily-armoured German ships and experiences the catastrophic loss of the Queen Mary and Indefagitable, and nearly loses his own flagship, Lion. Later, the battlecruiser Invincible will also be blown up after 90 seconds worth of salvos from the German battleline. The Grand Fleet would lose three battlecruisers, the HSF, just one.
Adm. John Jellicoe, CINC of the Grand Fleet, writes in his official dispatch:
The disturbing feature of the battle-cruiser action is the fact that five German battle-cruisers engaging six British vessels of this class, supported after the first twenty minutes, although at great range, by the fire of four battleships of the "Queen Elizabeth" class, were yet able to sink the "Queen Mary " and "Indefatigable." It is true that the enemy suffered very heavily later, and that one vessel, the "Lutzow," was undoubtably destroyed, but even so the result cannot be other than unpalatable.
The facts which contributed to the British losses were, first, the indifferent armour protection of our battle-cruisers, particularly as regards turret armour and deck plating, and, second, the disadvantage under which our vessels laboured in regard to the light. Of this there can be no question.
But it is also undoubted that the gunnery of the German battle-cruisers in the early stages was of a very high standard. They appeared to get on to their target and establish hitting within two or three minutes of opening fire in almost every case, and this at very long ranges of 18,000 yards. The German vessels appear to use some such system of fire as the Petravic method as the guns do not go off exactly together, and it unquestionably gives excellent results. The "spread" for both direction and elevation is very small and the rapidity of fire very great.
The lessons from Jutland are also going to be influenced by the restrictions of the Washington Naval Treaty in 1921; no more than 35,000 displacement tons, gun calibres capped at 16". So:
First, designs are going to tend towards the "all-or-nothing" approach for armor. Either a particular area is going to have armor, or it isn't. No more 1" plating for the deck and tapering armour from the stem to the stern. Battleships will have an armoured "box" protecting the most vital areas; usually from the first barbette to the last, and no more.
Secondly, as air power becomes more relevant in the inter-war years, speed and the ability to keep up with the carriers will take on new importance. No matter how much powder you pack into a gun, an airplane is going to be able to beat the range and firepower; the idea of lining up battleships and throwing explosives at one another is going to become obsolete very quickly. Improvements to oil-fired boilers and turbines instead of reciprocating engines are going push the speeds of the US Iowas into the 30 knots range.
Armaments are going to standardize; 15" inches for the UK and Germans in 4 double turrets. 16" in three triple turrets for the US. The US will eventually emphasize hitting power and speed over armor.
Finally, the necessity of central fire control and (mechanical) computerized fire direction cannot be overstated. The ships that had central FC scored more hits than the ships without. The ability for a fire solution to be calculated quickly and accurately from a central location meant more shells on or near target than previous designs considered.
Sources: Dreadnought and Castles at Sea by Robert K. Massie
Jane's Battleships of the 20th Century, Bernard Ireland, Tony Gibbons
US Battleship Operations in World War One; 1917-1919 by Jerry Jones