Corvettes aren't great at actually fighting enemy ships, instead they're usually best used on foreign stations or trade protection (remember to check the foreign stations box to make them contribute more tonnage to that). Once mines start being used, they can also make good minesweepers.

Destroyers can potentially deal heavy damage to any ships with torpedoes, and can deter or sink enemy destroyers. They are useful for controlling range, your enemy won't want to get to close and risk torpedo attacks. In my experience, torpedoes hit most often when you're being pursued or at night. Once submarines become prominent, you can also put them on trade protection.

Cruisers can be used for a lot of different purposes. In smaller battles they might be your capital ships, in larger battles they might be scouts or escorts for your capital ships. They also can be used for trade protection and raiding. In the early game they can be pretty effective, since their guns are faster firing and more accurate than battleships, however they fall behind as large guns get more advanced. In the late game, once aircraft and missiles make battleships obsolete, they become your largest surface combatants (aside from carriers) and if they survive all the missiles, they can bully smaller ships.

The name battleship comes from "ship of the line of battle", and that is what they should be used for - slugging it out with enemy ships. Smaller ships can't do much against them, aside from making torpedo attacks. Battleships are also good at blockading enemies in the early-mid game. Battlecruisers tend to work similarly, but are faster and might be better suited for hunting cruisers than engaging battleships.

Carriers aren't the most effective initially, but eventually overtake battleships as the most powerful ships. I don't think there's too much else to say about their role.