Before the introduction of aircraft carriers, battleships were the largest naval surface combatants. Modern battleships evolved from the dreadnought-type ship of the line after World War I.
A modern battleship is characterised by its large size, its full armour, and its armament. To be respected as a serious battleship rather than a mere heavy cruiser, a ship needs at least six 305 mm (12'') guns mounted in turrets. Usually, the guns were far heavier, though. The Bismarck had 380 mm guns, the Iowa class had 406 mm guns, and the main batteries of the Japanese failures mentioned below were a whopping 465 mm. Then, a decent battleship needs medium artillery (in the 127 mm range), and an enormous lot of tiny anti-aircraft guns. Usually, battleships had one or two small spotter seaplanes, too.
A ship with the same armament as a battleship but with less armour and thus a higher maximum speed is called a battlecruiser.
The first modern battleship --at least that's what my trusty Brockhaus encyclopedia tells me-- was the HMS Hood, later sunk by the Bismarck. Discussions about whether the Hood was a battlecruiser rather than a battleship keep turning up in military newsgroups, and in this node, too, apparently. The biggest battleships ever built, the Japanese Yamato and Musashi were huge failures.
Today, battleships are dead. None are active anymore, and there is no purpose left to them. They were used sporadically for shore bombardment, but even for that better solutions are now available. No matter how many people keep claiming it, there are no plans by the US Navy to reactivate any of the old Missouri class battleships. In fact, it would be impossible by now.