Cruisers
A cruiser is a multipurpose ship. During World War 2, there are three kind of cruisers: the heavy, the light and the antiaircraft ones.
The heavy cruiser (10 000 to 12 000 tons, with 10 or so 8 inches guns) is a small battleship. A little quicker (she was able to follow the carriers, which was not the case of most battleships) and with a handsome armor.
The light cruiser (6 000 to 8 000 tons, with 10 or so 6 inches guns) is less protected (the French and Italian light cruisers were especially badly protected). Her role is more akin to that of destroyer, as flotilla leader. She can operate at the same speed, with guns and torpedoes and can provide a solid complement of firepower.
The anti-aircraft cruiser is as big as a light cruiser (sometimes as a heavy cruiser), she has more but smaller guns (5 inches or less) and are able to lay a deadly fire on incoming enemy planes. They rarely operate independently or against surface ships, by lack of heavier guns.
Impy mentioned the battlecruiser. She is not really a cruiser but rather a battleship (as to the weight and the weaponry), in which protection is sacrificed for the need of speed. That kind of ship proved a failure during the battle of Jutland, during World War 1. Almost only remnants of that period appeared in the order of battle during World War 2. The newly built battlecruisers were in a fact either full blown battleships (such as the Scharnhorsts) or very heavy cruisers (such as the American Alaska). Both had an adequate protection.
A last word to mention the auxiliary cruisers (Armed Merchant Cruisers) which were merchant ships with a couple of guns. They were used either to attack isolated merchant ships (the German used a dozen of them), or to defend them (the British had much more of them).
Destroyers
A destroyer weight from 1000 to 3000 tons. She sails fast, up to 40 knots. She is the fastest ship in the fleet. Her weak point is the range. Whereas a battleship or a cruiser can sail 10 000 miles upward, the destroyer can hardly put more than 1000 or 2000 miles, a couple of days of sailing. That forced the task forces during the war to perform lengthy and dangerous fueling operations, where either fleet tankers or bigger ships (carriers, battleships) gave away part of their fuel to the guzzling destroyers.
The armament of a destroyer is about 6 guns, around 5 inches (The Germans built bigger destroyers with 6 inches guns, but they were not successful). Some World War 1 leftovers had guns of less than 4 inches. The most deadly weapon of the destroyer was the torpedo.
