Destroyers - DDs
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Light Cruisers and Torpedo Cruisers - CLs and CLTs
Torpedo Cruisers - CLTs
Heavy/Aviation Cruisers - CAs and CAVs
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Generally, when a certain ship subtype, such as CAV is mandatory, the designation is given sans brackets i.e. CAV, when either a CA or a CAV is allowed, a fleet composition will have the designation CA(V), denoting that aviation cruisers are allowed
Aviation Cruisers - CAVs
Battleships, Fast Battleships, and Aviation Battleships - BB, FBB, BBV
Fast Battleships - FBB
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Editor's note, yes, Kongous started out as Battlecruisers which would be designated CCs, but in Kancolle, they're often referred to as FBBs
Aviation Battleships - BBV
Light Carriers - CVL
Fleet Carriers and Armored Fleet Carriers - CVs and CVBs
Armored Fleet Carriers - CVBs
The good news is that there are only three armored carriers in the game