Frigate:
"Modern frigates are only related to earlier frigates by name. The term "frigate" passed out of use in the mid-19th century and was readopted during World War II by the British Royal Navy to describe a new type of anti-submarine escort vessel that was larger than a corvette, but smaller than a destroyer. The frigate was introduced to remedy some of the shortcomings inherent in the corvette design, namely limited armament, a hull form not suited to open ocean work, a single shaft which limited speed and manoeuverability, and a lack of range. The frigate was designed and built to the same mercantile construction standards (scantlings) as the corvette - allowing manufacture by yards unused to warship construction. The first frigates of the River class (1941) were essentially two sets of corvette machinery in one larger hull, armed with the latest Hedgehog anti-submarine weapon. The frigate possessed less offensive firepower and speed than a destroyer, but such qualities were not requisite in anti-submarine warfare (for instance, ASDIC sets did not operate effectively at speeds of over 20 knots). Rather, the frigate was an austere and weatherly vessel suitable for mass-construction and fitted with the latest innovations in anti-submarine warfare. As the frigate was intended purely for convoy duties, and not to deploy with the fleet, it had limited range and speed."
Destroyers:
"In naval terminology, a destroyer is a fast and maneuverable yet long-endurance warship intended to escort larger vessels in a fleet or battle group and defend them against smaller, short-range but powerful attackers (originally torpedo boats, later submarines and aircraft). Before World War II destroyers were light vessels without the endurance for unattended ocean operations; typically a number of destroyers and a single destroyer tender operated together."
"While destroyers would be able to hold off an attack by torpedo boats (which did not carry heavy guns), they would be easy targets for other warships when operating away from the fleet. Thus they were often given torpedoes of their own. While torpedo boats were short-ranged and cheap, destroyers had to operate as a screen for the fleet wherever it was. This required them to have the range of the ships they were protecting, requiring destroyers to be much larger than the boats they were designed to counter."
Light Cruisers:
"The term light cruiser was given a definition by the Washington Naval Treaty of 1921. The treaty, which sought to limit an arms race in warships, restricted the construction by nations of all large warships. Among its terms, cruisers could displace no more than 10,000 tons standard and light cruisers could be armed with guns of a calibre not exceeding 6.1 inches (155 mm) guns."
Heavy Cruisers:
"Heavy cruisers evolved from the light cruisers of World War I. The first heavy cruisers were the British Hawkins class (1915) of 9,750 tons that made 30 knots and were armed with seven 7.5 inch guns. These ships were directly descended from the preceding Birmingham (1911) group of Town class 5,440 ton light cruisers and at the time were referred to as "improved light cruisers". They remained the largest and most powerful cruisers in the world for several years."
"The Washington Naval Treaty of 1921 sought to stop an arms race in warships. It restricted the construction of warships of more than 10,000 tons standard displacement or with armament greater than 8-inch (203 mm) guns. The major naval powers then started to build cruisers up to those limits. These were usually between 9,000 and 10,000 tons and were typically armed with eight to ten 8-inch guns. In 1930 the Washington Naval Treaty was extended by the London Naval Treaty which split the treaty definition of a cruiser into Heavy Cruiser with guns larger than 6.1 inch (155 mm) and "light cruiser" with smaller caliber guns, creating the heavy cruiser as a category of warship. The upper limit of 10,000 tons displacement still applied to both"
"In the 1930s several navies began to secretly flout the tonnage limits. The Japanese who were planning on withdrawing from the treaty built the Mogami class with a displacement of over 12,000 tons. She was designed so that her five 6.1-inch gun triple turrets could be replaced with twin 8-inch gun turrets and the ships of the class were rearmed in that way shortly before World War II. Japan withdrew from the Washington Naval Treaty in 1936, after which there was no effective regulation."
Battle Cruisers:
"Battlecruisers were large warships of the first half of the 20th century first introduced by the British Royal Navy. They evolved from armoured cruisers and in terms of ship classification they occupy a grey area between cruisers and battleships. Generally, battlecruisers were similar in layout and armament to battleships but with significantly less armour allowing for gains in speed.
"The German pocket battleships (German

anzerschiffe - armored ship: Deutschland, Admiral Scheer, and Admiral Graf Spee), built to meet the 10,000 ton displacement limit of the Treaty of Versailles, were another attempt at a cruiser-battleship concept. Rather than construct a lightweight battleship which sacrificed protection in order to attain high speed, the pocket battleships were relatively small vessels with only six 11 inch (280 mm) guns — essentially large heavy cruisers. They attained fairly high speeds of 26 knots (52 km/h), and reasonable protection, while (allegedly!) staying close to the displacement limit, by using welded rather than riveted construction, triple main armament turrets, and replacing the normal steam turbine power with a pair of massive 9 cylinder diesel engines driving each propeller shaft (an ironic reversion from turbine to reciprocating engines). They were later reclassified as "heavy cruisers", having heavier guns and armour than regular heavy cruisers at the cost of speed. Unfortunately, they were outclassed by British WW1-era true battlecruisers in speed, weaponry, and protection. (They in fact had basic cruiser armour, except for the turrets.)
Two more ships were built later in the 1930s, the Scharnhorst and Gneisenau, which were considerably more powerful. At 38,900 tons full load they were somewhat larger than the French Dunkerque class and very well armoured. As built, they had three triple 11-inch gun turrets, but the mounts were designed to accept twin 15-inch turrets (six guns total) when enough became available. However, circumstances and the fates of the two ships - Scharnhorst sunk by gunfire, Gneisenau heavily damaged by bombs and her repair sacrificed to higher priorities - meant that this plan was abandoned. At the time, treaty requirements allowed the production of 12+ inch guns at 1 a year, which along with the very time consuming production of naval guns, kept these two ships with 11-inch guns. The Royal Navy categorised them as battlecruisers since they followed the Imperial German Navy design lineage of trading off gun size for protection and speed. The German Navy nonetheless categorised them as battleships."
Although technical specifications varied, all battlecruisers shared a similar role specification. They were designed to hunt down and outgun smaller warships (or merchant ships in the case of the pocket battleships), and outrun larger warships that they could not outgun."