Naval guns are generally considered to be inferior to aircraft and missiles for the simple reason that they are. The longest range ever achieved with a British naval gun was at a range of 51,000 yards. This was with a gun in a special high-elevation land-based mount and using supercharges that were never issued to ships at sea. If we only count guns actually mounted on ships, then the longest range achieved was the 40,500 yards that the 18in Mark I was capable of when mounted on the monitors General Wolfe and Lord Clive. General Wolfe scored the longest-ranged hit ever achieved by a British warship, at a range of 36,000 yards - but this was against a bridge in Flanders. The longest-ranged hits against other ships were at ranges of ~26,000 yards, scored by Scharnhorst against HMS Glorious and by Warspite against Giulio Cesare. Aircraft and missiles can easily outrange this. The first carrier strike aircraft, the Sopwith Cuckoo of 1917-8, had a range of about 160 nautical miles (325,000 yards), six times as far as the best possible battleship could and twelve times as far as a realistic battleship could. As technology improved, the superiority of the aircraft became more clear. It could easily outrange a battleship, sinking it before it could get into range of a carrier.

For missiles, this is less true; the first surface-to-surface missiles had comparable range to the longest ranges achieved by battleship guns. Other factors stood in their favour. Firstly, missiles could deliver much more explosive much more accurately than a battleship. Shells need thick walls to withstand the forces of being fired out of a gun, reducing the space available for a bursting charge; missiles (and aircraft bombs or torpedoes) do not face such problems. The heaviest battleship shells ever used, those used by the Japanese 18.1in (46cm) gun, had a bursting charge of 74.6 kg in the armour-piercing variant, or 136 kg in the high-explosive variant. The comparatively small Martel missile, used by British and French aircraft in the 1970s and 80s, had a 150 kg warhead; large surface-to-surface missiles like the Soviet SS-N-2 'Styx' have warheads of up to 450 kg or higher. It is also much easier to fit a nuclear warhead into a missile or bomb due to the same factors.

Secondly, missiles and aircraft are equally accurate throughout their ranges. Missiles have guidance systems, typically radar, which can redirect their course in flight. Aircraft, through their pilots, can also change course after being launched. Shells, though, cannot. They rely on calculations made before they are fired; this means that evasive action by the target or small errors in measurements of target range, course or speed can cause the shells to miss. Naval guns, especially if the barrels are worn, are also not inherently the most precise things; firing at a particular point will result in a wide spread of shells falling around that point. Missiles and aircraft do not suffer from these problems to the same extent; they are much more likely to hit their targets as they are guided in flight.

Finally, large-calibre naval guns are large, expensive and complex things. To construct them requires tens of tons of steel to undergo a number of complex forging operations; this in turn needs a large amount of infrastructure and vast, expensive machine tools. The same is true of the turrets they fit into, which are some of the most complex things ever put to sea. Missiles and aircraft are generally simpler to produce, relying more on smaller, easily produced components. To be effective, a large-calibre naval gun needs to be put on a ship that is large enough to support it, to contain the magazines to feed it, the rangefinders and fire control computers to make it work, and so on. Such ships are hugely expensive. A missile can easily be put onto a small, cheap ship; it just needs a radar and a launch rail or tube. Aircraft are also comparatively cheap to operate, and can operate from a relatively cheap ship.

All of this explains why the large-calibre naval gun has declined in use. Aircraft and missiles, while they can be stopped, have longer ranges, are more accurate, and do more damage than a battleship shell. It should also be noted that the battleship shell can be stopped, by sinking the battleship before it gets into range; this can easily be done with aircraft or missiles. It was clear in the years after WWII that this was true; the losses of Prince of Wales and Repulse to land-based aircraft in 1941, and the losses of Yamato and Musashi to carrier-based aircraft in 1944-5 are the main examples that are pointed to. Navies began to place their battleships in reserve immediately after the war in favour of keeping carriers in operation; the RN sold off its battleships in the mid 1950s, with the last, Vanguard, leaving service in 1960. Cruisers, which were more capable of fighting off aircraft when fitted with a main armament of fast-firing guns, did stick around to some extent. However, as missile-armed ships became more prominent, the gun-armed cruisers were soon sold off or reduced to reserve and secondary roles.

Naval guns are not completely obsolete, though. In the 1970s, the RN designed a new class of general-purpose/ASW frigates, the Type 22. This had Exocet surface-to-surface missiles, Sea Wolf surface-to-air missiles for point defence against aircraft and missiles, and a helicopter and torpedoes for anti-submarine work. They were not given a gun, as it was felt that they would not need it. In 1982, two Type 22s, Brilliant and Broadsword, were part of the British Task Force that liberated the Falklands. Here, their lack of a gun was seen as detrimental. The Royal Navy needed ships armed with guns to provide fire support to troops ashore or to shell airfields and other strategic targets. The lack of gun on the Type 22s meant that older ships with less effective SAMs were sent to do this duty, leading to a number of losses to Argentine aircraft. The RN had no missiles that could fire at targets ashore, and while it did have aircraft, it did not have enough to hit every target that needed to be hit. The gun was also found to be a useful standby for defence against aircraft that had managed to evade the ships' SAMs. When a new batch of Type 22s was ordered following the war, they were redesigned to add the RN's standard 4.5in gun.

Naval artillery is not entirely obsolete, though it's role has certainly diminished since the Dreadnought era.

In the US Navy, the Arleigh Burke class destroyers (the most common American surface combatant) all carry a 127mm gun. The Zumwalt class, which was intended to replace the Arleigh Burke class before being cancelled due to cost overruns, carries what's called the Advanced Gun System (AGS), which is exactly what it sounds like. As a side effect of the Zumwalt cancellation, however, there's no ammunition for the AGS and it can't be used in practice. Other American surface combatants also carry artillery.

The decline of big naval guns can be mostly clearly traced to WWII. Guns have a limited range -- the longest distance naval hit from artillery in history was at a distance of 15 miles. With the development of the aircraft carrier, it became increasingly unlikely that ships would get that close to one another. Famously, during the Battle of the Coral Sea (1942), the opposing naval forces never even fired at one another; aircraft did the fighting.

Subsequent battles in the Pacific also took place entirely "over the horizon," and it became clear that the days of naval forces engaging at close range were numbered. Today, the US Army has a gun that can hit a target reliably at a little over 40 miles. The combat radius for carrier-based aircraft is in the hundreds of miles (and further for missiles), so a naval engagement is likely going to be over before anyone gets that close.

Although big guns ceased to be effectively in a ship on ship engagement in the WWII era, they remained useful for shore bombardment. Battleships proved effective for the United States in attacking shore defenses or points immediately inland both in the Pacific and as part of the Normandy campaign. Nonetheless, the ships were considered outdated and the US deactivated its battleships immediately after the war with the one exception of the USS Missouri. A number of ships were reactivated for various conflicts, mostly with the shore bombardment role in mind, and there was a brief resurgence of the battleship as part of Ronald Reagan's push for a 600-ship navy with WWII era battleships re-entering the service. The last battleship gunfire support operations came during Operation Desert Storm, and the last decommissioned battleships were turned into museums.

But, as I said at the outset, the end of the big gun battleship was not the end of naval guns and other US surface combatants are still capable of fulfilling the naval gunfire support role. There was a considerable debate about this in the 1990s and agreement at the time that the capability was still relevant. Naval gunfire support has played a minor role in US military operations in this century but is still used (e.g., during the invasion of Iraq and the operation in Libya).

I've been focusing my answer here on the US Navy. If we look globally, where the average navy operates mostly in a coastal defense role, naval artillery still plays a central role. For patrol vessels, guns are still the main system.