Battleships have become obsolete because in the modern threat environment they are not as capable at dealing damage as other possible options. Aircraft were the battleship's main competitor at dealing damage immediately after WW2. Compared to a battleship, aircraft could deliver a heavier strike, over a longer distance, and much quicker. For example, for a target 300 miles away, it would take the typical fast battleship 10 hours at full speed to reach it. Aircraft would take a tenth of the time. The typical carrier aircraft cost tens of thousands of pounds, while a battleship cost closer to ten million. Hundreds of aircraft could be produced, maintained and flown for the cost of a battleship, and together those aircraft packed a bigger punch. Aircraft were more versatile than a battleship, and could project power over a much larger radius - including inland. A battleship can really only contest only the sea within range of its guns, while aircraft can contest anywhere they can fly to.

Battleships also represent a major liability - in a threat environment where everything is vulnerable, having a single, large, expensive target making up the majority of your combat power is foolish. If you lose a single aircraft, or small warship, then you have lost comparatively less than if you lose a battleship. Modern weaponry, especially nuclear weapons (as demonstrated at Bikini Atoll) make battleships just as vulnerable as any other ship or aircraft.

The reason why Battleships are now obsolete is down to its range and ability to attack.

During WWI, Battleships were the mainstay of all fleets during the war. They decided battles, and whoever had the better designed battleships or better tactics would win. Britain had the larger fleet during the First World War, however their ships were not as well designed as Germany's. This was best shown during the Battle of Jutland, Where Britain had 151 ships in the Grand Fleet, while Germany only had 99, with only 16 battleships compared to Britain's 28, and yet Britain suffered worse losses during the battle due to bad design.

However, this doesn't answer the question, just some background.

During WWI, Carrier warfare was beginning to be developed. It started with battleships carrying small scout planes with the ability to land on water (however mainly it lead to a crash landing and the plane being fished out). For example, HMS Engadine was a converted Cross-Channel Packet ship which was converted to carry planes. She participated in Jutland with only four planes. One of said planes did manage to spot three German cruisers and five destroyers, which was the first use of a heavier-than-air aircraft had carried out a reconnaissance of an enemy fleet in action.

This was continued, with the first "true" carrier being the HMS Ark Royal, which a flat flight deck.

Now, moving onto WWII. Battleships still played a major part in this war, with them still being the mainstay of nations fleets, however, as shown early on in the war with the Bismarck, battleships which are insufficiently protected with AA are extremely vulnerable to air attack, even with antiquated bi-planes attacking a modern battleship they still managed to cripple the Bismarck which allowed the British fleet to finish her off.

Battles were increasingly being decided by a nations carrier air power. The Japanese strike on Pearl Harbour was to cripple the American navy, and as such they expected to cripple the carrier fleet, however the carriers were not at Pearl Harbour during the raid. If they were, it could have had dire consequences on the war, potentially prolonging it and as such battles such as Midway would have been entirely different.

That nicely ties us onto the Battle of Midway. Both Japanese and American fleets had carriers, 4 and 3 respectively, with 2 light carriers which did not participate for the American fleet. These carrier borne planes were a decisive part of the battle, both reporting enemy fleet locations and striking enemy ships. The Japanese loss of 4 carriers was crippling for Japan, as they had very limited resources and were unable to replace said carriers during the war.

The Battleship, especially today would be quite useless as it relies on large guns and long range, however with modern missile technology, long range nuclear submarines and long range jet fighters/bombers, they would be sitting ducks when attacked. Destroyers are the backbone of fleets now, as they are reliable and versatile.

I hope this answers your questions.

Also, first post. Yay.

A related question for anyone interested: Why have carriers not become obsolete due to the significant advance of aa capabilities?