I found this list very informative. From what I understand, this is how the modern US Navy classifies their ships. So the Starfleet Command Constituation FCA would be a Heavy Gun Cruiser according to this list.
AB Crane Ship
ABD Advanced Base Dock
ABSD Advanced Base Sectional Dock
AC Collier
ACS Auxiliary Crane Ship
AD Destroyer Tender
AE Ammunition Ship
AEFS Fleet Replenishment Ship
AF Stores Ship AFD Mobile Floating Dry Dock
AFS Fleet Stores Ship
AG Miscellaneous Auxiliary
AGC Amphibious Force Flagship
AGDS Launch Area Support Ship
AGF Command Ship
AGL Lighthouse Tender (Coast Guard)
AGM Missile Range Instrumentation Ship
AGP Motor Torpedo Boat Tender
AGOR Oceanographic Research Ship
AGOS Ocean Surveillance Ship
AGS Surveying Ship
AGSS Auxiliary Research/Experimental Submarine
AGTR Auxiliary Technical Reconaissance Ship
AH Hospital Ship
AK General Cargo Ship
AKA Attack Cargo Ship
AKD Deck-Hold Cargo Ship
AKN Net Cargo Ship
AKR Vehicle Cargo Ship
AKS General Stores Carho Ship
AKV Aircraft Ferry
AL Lightship (Coast Guard)
AM Fleet Minesweeper (steel)
AMB Base Minesweeper
AMC Coastal Minesweeper (wooden)
AN Net Laying Ship
AO Oiler
AOE Fast Combat Support Ship
AOG Gasoline Carrier / Tanker
AOR Replenishment Oiler AOT Transport Oiler / Tanker
AP Troop Ship, non-landing
APA Attack Transport Ship
APB Barrack Ship, Self-Propelled
APC Small Coastal Transport
APD Fast Destroyer Transport
APF Adminstration Flagship
APG Supporting Gunnery Ship
APH Transport, Evacuation of Wounded
APL Auxiliary Personnel Light / Labour Transport Ship
APM Mechanized Artillery Transport
APN Non-Mechanized Artillery Transport
APR Rescue Transport
APS Transport Submarine
APSS Vehicle Transport Submarine
AR Repair Ship
ARB Repair Ship, Battle Damage
ARC Cable Repair/Laying Ship
ARD Floating Dry Dock
ARD Repair Dock, Concrete
ARG Repair Ship, Landing Craft / Repair Ship, Internal Combustion Engine
ARH Repair Ship, Hull
ARL Repair Ship, Landing Craft
ARM Repair Ship, Heavy Machinery
ARS Salvage Vessel
ARV Aviation Repair Vessel
AS Submarine Tender
ASR Surmarine Rescue Vessel
ASSP Auxiliary Transport Submarine
AT Tug
ATA Auxiliary Tug
ATF Oceangoing Fleet Tug
ATO Old Tug ATR Recue Tug
ATS Salvage & Rescue Ship
AV Seaplane Tender
AVC Catapult Lighter
AVD Seaplane Tender (Destroyer)
AVP Seaplane Tender (Small)
AVR Aircraft Rescue Vessel
AVS Aviation Supply Ship
AVT Auxiliary Aircraft Transport
AW Distilling Ship
AWK Water Tanker
AZ Airship Tender (Lighter-Than-Air)
BB Battleship
BC Battlecruiser
BCGN Nuclear-powered Guided Missile Battlecruiser
BRDM Floating Dry Dock
CA Heavy Gun Cruiser
CAG Guided Missile Heavy Cruiser
CB Large Cruiser
CC Tactical Command Ship
CG Guided Missile Cruiser
CGN Nuclear-powered Guided Missile Cruiser
CHG Helicopter Cruiser with Guided Missiles
CL Light Cruiser
CLAA Light Anti-Aircraft Cruiser
CLC Task Fleet Command Ship
CLG Guided Missile Light Cruiser
CLGN Guided Missile Light Cruiser, Nuclear-powered
CLK Hunter-Killer Cruiser
CM Minelayer CMC Minelayer, Coastal
CONT Container Ship
CV Aircraft Carrier
CVA Attack aircraft carrier
CVAN Attack aircraft carrier, nuclear powered.
CVB Large Aircraft Carrier
CVE Escort aircraft carrier.
CVH Helicopter Carrier
CVHA Assault Helicopter Transport
CVHE Escort Helicopter Aircraft Carrier.
CVHG Helicopter Carrier with guided missile armament.
CVL Light aircraft carrier.
CVN Nuclear-powered aircraft carrier.
CVS Support (ASW) Aircraft Carrier.
CVT Training Aircraft Carrier
CVU Utility Aircraft Carrier
DD Destroyer
DDE Escort Destroyer
DDG Guided Missile Destroyer
DDH Helicopter Destroyer
DDK Hunter-Killer Destroyer
DDR Radar Picket Destroyer
DE Escort Destroyer DEC Control Escort Ship
DEG Guided Missile Escort Ship
DER Radar Picket Escort Ship
DL Frigate
DLG Guided Missile Frigate
DLGH Guided Missile Frigate, Helicopter
DLGN Guided Missile Frigate, Nuclear-powered
DM Light Minelayer (Former Destroyers)
DMS Minesweeper, Fast (Former Destroyers)
EDD Experimental Destroyer
EDE Experimental Escort Ship
FF Frigate
FFG Guided Missile Frigate
FFGH Guided Missile Frigate, Helicopter
FFH Frigate, Helicopter
FFL Corvette
IFS Inshore Fire Support Ship
IX Unclassified Auxiliary
IXS Unclassified Auxiliary Submarine
LASS Launch Area Support Ship
LCAC Landing Craft, Air Cushioned
LCC Command Ship for Amphibious Operations
LCI Landing Craft, Infantry LCM Landing Craft, Mechanized units
LCP Landing Craft, Personnel
LCT Landing Craft, Tank
LCU Landing Craft, Utility
LCVP Landing Craft, Vehicles and Personnel
LHA Helicopter Assault Ship
LHD Amphibious Assault Ship
LKA Amphibious Cargo Ship
LPA Amphibious Transport
LPD Amphibious Transport Dock
LPH Amphibious Assault Ship
LPSS Amphibious Transport Submarine
LS Landing Ship
LSD Landing Ship, Dock
LSI Landing Ship, Infantry
LSM Landing Ship, Medium
LSMR Landing Ship, Medium, Rockets
LST Landing Ship, Tank LSU Landing Ship, Utility
LVT Landing Vehicle, Tracked
MCM Mine Countermeasures Ship
MHC Coastal Mine Hunter
ML Minelayer
MS Minesweeper
MSB Minesweeping Boat
MSH Minesweeper,
MSO Minesweeper, Ocean
MV Merchant Vessel
PBR Patrol Boat, River
PC Patrol Corvette
PCE Escort Patrol Corvette
PCS Patrol Corvette, Minesweeping
PF Patrol Frigate
PG Patrol Gunboat
PGH Patrol Gunboat, Hydrofoil
PGM Patrol Gunboat, Motor
PHM Missile-armed Patrol Hydrofoil.
PM Patrol Missile Boat
PT Torpedo Boat
PTG Torpedo Boat, guided missile armament
PTM Missile-armed Torpedo Boat
PY Patrol Vessel (Converted Yacht)
PYC Coastal Patrol Vessel (Converted Yacht)
SC Submarine Chaser (Wooden)
SM Minelayer Submarine
SS Submarine
SSA Cargo Submarine
SSB Ballistic Missile Submarine
SSBN Nuclear-powered Ballistic Missile Submarine
SSG Guided Missile Submarine
SSGN Nuclear-powered Guided Missile Submarine
SSK Hunter-Killer Submarine
SSKN Hunter-Killer Submarine, nuclear powered
SSN Nuclear-powered (attack) Submarine
SSO Oiler Submarine
SSP Vehicle Transport Submarine
SSR Radar Picket Submarine
SSRN Radar Picket Submarine, nuclear powered
TANK Tanker
TRAN Transport
YA Ash Lighter
YAG District Auxiliary, Miscellaneous
YC Open Lighter
YCF Car Float (Railroad)
YCK Open Cargo Lighter
YCV Aircraft Transportation Lighter
YD Floating Derrick
YDG Degaussing Vessel
YDT Diving Tender
YE Ammunition Tender
YF Covered Lighter
YFB Ferry Boat / Launch
YFD Floating Dry Dock
YF Torpedo Transportation Lighter
YG Garbage Lighter
YH Ambulance Boat
YHB House Boat
YHT Heating Scow
YLA Open Landing Lighter
YM Dredge
YMS Motor Mine Sweeper
YN Net Tender
YNG Gate Vessel
YNT Net Tender, Tuc Class
YO Fuel Oil Barge
YOG Gasoline Barge
YOS Oil Storage Barge
YP District Patrol Craft
YPD Pile Driver (Floating)
YPK Pontoon Storage Barge
YR Floating Workshop
YRD Floating Workshop, Drydock
YS Stevedoring Barge
YSD Seaplane Wrecking Derrick
YSP Salvage Pontoon
YSR Sludge Removal Barge
YTB Large Harbour Tug
YTL Little Harbour Tug
YTM Medium Harbour Tug
YTT Torpedo Testing Barge
YW Water Barge
LCC Amphibious Command Ship
LHA Amphibious Assault Ship
LHD Amphibious Assault Ship
LKA Amphibious Cargo Ship
LPA Amphibious Personnel Transport
LPD Amphibious Transport Dock
LPH Amphibious Assault Helicopter Carrier
LSD Dock Landing Ship
LST Tank Landing Ship
AD Destroyer Tender
AE Ammunition Ship
AF Refrigerated Stores Ship
AFS Combat Stores Ship
AG Miscellaneous Auxiliary
AGF Flagship
AGOR Oceanographic Research Ship
AGS Surveying Ship
AK Cargo Ship
AO Fleet Oiler
AOE Fast Combat Support Ship
AOR Replenishment Oiler
AR Repair Ship
ARS Salvage Ship
AS Submarine Tender
ATF Fleet Tug
AV Seaplane Tender
AVT Aircraft Transport, later Training Aircraft Carrier
CM Minelayer (derived from Cruiser-Minelayer)
DM Destroyer Minelayer (or light minelayer or high-speed minelayer)
DMS High Speed Minesweeper (derived from Destroyer-Minesweeper)
MCM Mine Countermeasures Ship
MCS Mine Countermeasures Support Ship
MF Fleet Minelayer
MHC Coastal Minehunter
MSB Minesweeping Boat
MSC Coastal Minesweeper
MSO Ocean Minesweeper
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