I know old battleships were designated BB. Destroyers were DD (I think).
Having trouble with carriers.

If I'm not mistaken, the C stands for carrier and the A stands for Attack.

What's the V  ?

fixed wing

On this page: link

ETA: CA was already used for Armored, V was then used for Aviation.

Base Designations
Suffixes are added to these to create complete designations.
Where no suffix is used, the initial letter is used twice (i.e. DD).
A suffix is often added to an existing two- or three-letter abbreviation at a later date (i.e. DDG, AGF, SSGN).

A –– Auxiliaries
B –– Battleship/battlecruiser
C –– Cruiser
CV –– Aircraft Carrier (originally a cruiser derivative)
D –– Destroyer
F –– Frigate
L –– Ampbibious ships (derived from "L" for "Landing")
M –– Mine craft
P –– Patrol craft
S –– Submarine
Y –– Yard craft

Suffixes
These and others are added to base designations to create complete designations.
Many suffixes (i.e. M) have many meanings; only the most common modern-usage meanings are listed.

A –– Attack or Auxiliary
B –– Ballistic Missile
C –– Command/Control
D –– Destroyer or Dock
E –– Escort (combatants); Ammunition (auxiliaries)
F –– Fast or Fleet or Refrigerated Stores or Fire Support
G –– Guided Missile (AAW) (large vessels) or Gun (small vessels)
H –– Helicopter or Hospital
K –– Cargo
L –– Light
M –– Missile (surface-to-surface) or Missile Tracking
N –– Nuclear Power
O –– Oil (tanker)
P –– Personnel/Troop Transport
R –– Repair or Replenishment or Roll-On/Roll-Off (cargo)
S –– Surveying or Salvage or Submarine
T –– Tug or Transport
U –– Utility
V –– Heavier-Than-Air (later fixed wing) Aircraft or Vehicle (transport)
W –– Water

CV –– Aircraft Carrier
CVA –– Attack Aircraft Carrier
CVAN –– Nuclear Powered Attack Aircraft Carrier
CVE –– Escort Aircraft Carrier
CVL –– Light Aircraft Carrier
CVN –– Nuclear Powered Aircraft Carrier
CVS –– ASW Aircraft Carrier (sometimes Seaplane Carrier, esp. WWII)

I always thought CV meant Carrier Vessel

Too bad that code doesn't allow correct designations such as CHOAD, FUBAR, etc.  

V-fixed wing

H- Rotary wing

CVN

LPH

VVVVVVVVVVVVRRRRRRRRRRRRRRRRRRRRROOOOOOOOOOOOOOOOOOOOOOOOOOMMMMMMMMMMMMMMMMMMMMM!

V is for victory!

but modern carries have both fixed and rotary wing, shouldn't they be CVHN?

Originally, V was for "Heavier than Air."

So VF-1 was a Heavier than Air, Fighter Squadron.
So VS-1 was a Heavier than Air, Scout Bomber

They are a Carrier, Heavier than Air, Nuclear.  CVN.

An amphib helo carrier would be Landing Craft Carrier (LH) Assualt.  LHA.  The helicopters are landing craft for Marines.

A carrier built for operations at sea for naval Helos would be Carrier, Heavier than Air, Helo or CVH.  There were CVHs for awhile, I think in the 1950s and 1960s.

The L in LPH is for "Landing Platform, Helicopter" meaning a carrier, employed as an amphib, without a well deck or other amphib qualities.

It's kind of neat that the USS Eisenhower and Nimitz were originally comissioned CVAN.  A = Attack.
So was the Enterprise IIRC.

M= MARINE

I actually have a pre-75 CVAN-75 Patch around here...

The Attack was to differentiate them from CVS anti-submarine carriers.

Focuses on primary role.

V- Fixed wing squadrons w/ some helo support for SAR and Vert Rep

H- MEU chop.  mostly swing wings with a few Lawn Darts thrown in supporting a bunch of killing machines

Fixed wing aircraft.

Like you have a VF for a fighter squadron, or VA for an attack squadron.

CV essentially stands for aircraft carrier.  It is their prefix.  C was already in use by cruisers.  Adding a V made sense since that was the designator for ships that had something to do with aviation.  A seaplane tender, for example, if I remember correctly, would be an AVP.  In this context it has nothing to do with fixed wing versus rotary wing.  Rotary wing aircraft did not even exist back when the dsignator was created.  You can actually have a helicopter carrier, which I have seen labeled a "CVH" when describing foreign ships of the type.  The only ships of this type I can think of that are currently operational are the new Japanese ones, although the Japanese, to avoid the restrictions in the WWII peace treaty, call them helicopter destroyers, or DDH.