There's quite a few terms, each with a long history behind them. Most of them come from the tactical role each type of ship was designed to fulfil.

The term battleship is a contraction of an 18th Century term, 'line-of-battle ship', and first used in the 1790s. This was used to denote the larger ships in the fleet, the ones capable of fighting in a fleet action. Because of the difficulty of commanding a fleet in a time before radio, and when battlefields were choked by gunsmoke, battles were typically fought by ships in line astern. This ensured that an admiral could effectively maintain control over his fleet, as all ships would follow the ship in front. Ships that were though capable of forming part of this line were called line-of-battle ships, though they were more commonly referred to by their rate (essentially how many guns they had, with a first rate having ~100 guns and a 3rd rate having typically 74). As warship design progressed during the 19th Century, with steel construction and turrets making an appearance, the difference between the rates vanished. Instead, the ships came to be called battleships, as line-of-battle ship was much too long for typical use.

Cruiser is yet another term that dates back to the Age of Sail. In this period, a cruiser was a small ship, a frigate, sloop, corvette or brig, detailed to 'cruise' - to sail around and protect trade or harry the enemy's merchant ships. As with the battleship, these ships would typically be referred to by the classifications common at the time. Sloops and brigs were so named because of their sail plan, while corvette comes from the French (itself borrowing from Dutch) for 'small ship' and frigate comes from an Italian word for a small fast ship. Once again, the design choices forced by the new technologies of the 19th Century forced a homogenisation of these varied types, into a single new one, the cruiser. This was a ship designed to scout for the fleet, protect trade, and raid the opponent's trade, able to cruise alone.

There were several sub-classes of cruiser. The protected cruiser came first, so named because it was given a protective, curving, armour deck, in contrast to the earlier unarmoured cruisers. This was then succeeded by the armoured cruiser, which also received an armour belt like those fitted to contemporary battleships. The term battlecruiser was originally used in 1893 by a British officer, Samuel Long, to refer to newly built large armoured cruisers, that appeared capable of carrying out the duties of the battleship. However, it is more commonly used to refer to large, well-armed cruisers inspired by the Invincible of 1907, which again, were seen as being able to operate as a major part of a fleet should the occasion require it. Light and heavy cruisers were the result of the naval treaties of the 1920s and 30s, which sought to limit the size and power of cruiser fleets - essentially, a light cruiser is a cruiser armed with 6in guns, while a heavy cruiser was armed with 8in guns.

The destroyer, meanwhile, was a 19th Century invention. Following the invention of the Whitehead Torpedo in 1866-68, navies began to build a variety of fast craft to carry them into combat. Starting with HMS Lightning of 1873, there was a rapid profusion of torpedo-carrying craft, given the general name of 'torpedo boats'. These posed a major threat to the battleships and cruisers common at the time, and much thought was given to how to protect such ships against attacks by torpedo boats. The general conclusion was that an escort ship was needed. In the 1880s, the RN experimented with 'torpedo gunboats' (TGBs) - essentially small cruisers for fleet protection. However, it quickly became clear that the TGBs were too large, expensive and slow to adequately protect the fleet from torpedo boats. Instead, a new class of ship was designed, with the first entering service in the 1890s. These were called 'torpedo boat destroyers', and were essentially enlarged torpedo boats with a heavier gun armament. Once again, the full name was found to be cumbersome, and so it was shortened to 'destroyer', though the type retained its role as a fleet escort.

There's a few other ship classes you've not asked about directly. The aircraft carrier is self-explanatory (it carries aircraft), but the classification for the type, CV, used by the USN is not. It comes from the term 'cruiser voler', from the French word for 'flight' - essentially, it means aviation cruiser. The classification isn't universal, with the RN preferring to use 'R' for carriers (they would have used 'A', but auxiliary vessels were using that letter first). The modern frigate and corvette bear little resemblance to their earlier namesakes. In the late 1930s, the RN was looking for a classification for its new Flower-class escorts, and chose corvette thanks to its association with naval history. When they built the larger River-class a few years later, frigate was chosen to describe the new ships - a frigate was essentially a larger corvette, and the term again had a close association with the history of the RN.

Ship typing is always a little arbitrary -- cf the new Zumwalt class of "destroyers" for the USN that, at 600 feet and 14,500 tons, are about the size of most WWII heavy cruisers. But the basic breakdown you'd get from the period between say 1906 and WWII is:

A "battleship" is a designation for a warship that is heavily armored and carries large guns, designed to be the capital ship of a navy. The term "battleship" descends from "line of battle ship," a ship able to stand in the line of battle, though at the time of sailing navies those ships would usually be referred to by their rates. The first use of "battleship" recorded by the OED is in 1794, in a poetic context. The term starts being applied to what we think of as a battleship today around the time of HMS Dreadnought, the first all-big-gun battleship, and continues to refer to that type of ship. There are no battleships currently in commission in the world's navies; the U.S. has preserved several older battleships. This is the USS Missouri, preserved at Pearl Harbor adjacent to the Arizona memorial. No navies operate battleships now, because aircraft carriers and ships and submarines carrying torpedoes and antiship missiles are able to project power over the horizon.

Battleships would have had a main armament of guns that would be 12" or larger (that is, the diameter of the shell would be 12"), ranging up to the 18.1 inch (46 cm) main guns of the superbattleships Yamato and * Musashi.*

A battlecruiser is a ship that's armed like a battleship, but armored like a cruiser. Design theory for battleships is complicated, but the very basic idea is that a battleship design should be proof against its own caliber of guns -- that is, the armor a battleship carries should be able to keep enemy shellfire out at normal fighting ranges. Battlecruisers sacrifice heavy armor for speed, and were theoretically supposed to be scouts for a battle fleet, but when they engaged battleships, the results were disastrous.

A cruiser is a warship designed to be fast and long-ranged, to both harass enemy commerce and to serve in a scouting role for the battle fleet. Cruisers came in "heavy" and "light" varieties, which were formally designated in the 1922 Washington and 1930 London naval treaties, and cruiser roles were varied among different navies (the Japanese, for example, built heavy torpedo armament into their cruisers and even built some cruisers with main armament of torpedoes, with the goal of attritting the American fleet as it moved across the Pacific). The London treaty defined heavy cruisers as ships carrying larger than 6.1" armament, with light cruisers carrying less than 6.1" guns; both were theoretically supposed to be limited to 10,000 tons displacement, but in practice navies ignored those treaty limits.

A destroyer is a small, fast warship that fills a variety of roles, but one of their most important roles in WWII was antisubmarine warfare. Destroyers would carry small, fast-firing main guns, usually had torpedo tubes, and would be armed with a variety of antisubmarine weapons, such as depth charges or anti-sub mortars. They would generally not have armor plating and could be quite small -- Yamato's main turrets were heavier than many of her escorting destroyers!