Here's a basic, universal guide for ship classes, based upon their real-life naval counterparts and other science-fiction spaceship classifications. These can be altered and modified to fit their racial versions, but this will mostly cover it. Please notify me if you take issue with any of these classes.
Small Spacecraft (Single or double-seat, swarm attack craft)
Fighters/Bombers: Small, general-purpose spacecraft used for engaging other small vessels or attacking larger ships in swarms.
Drones: Generally referring to any small, unmanned ship, usually under the control of a carrier-type vessel. Standard drones operate similarly to fighters, although advanced drone technology can serve various purposes and can range in size anywhere from a microship to a capital ship.
Reconnaissance: Usually just "recon" for short, these vessels are the eyes and ears of most navys. Their offensive weaponry and defensive capabilities are limited, so their ships are outfitted to be extremely maneuverable and fast to escape destruction. Usually, recon ships are ignored by most vessels because chasing after one is akin to chasing after a dragonfly. Nearly impossible, with little tactical reward. As a result, their pilots tend to be hot-shot dare devils with a need for speed, but if a fleet needs to know what's-what, then they use their recon ships to figure it out. The only ships that dare attempt such prey are some varieties of drones, gunmen on larger vessels without other targets, and the stealthy predators.
Gunships: Fairly large for small vessels, gunships are slower and better armored than their counterparts, and can easily get in and out of tight spots in a hurry. They are capable of operating both in and out of atmosphere, and are commonly used for boarding, troop transport, and close-air support.
Small to Mid-Size Vessels (Fast, crewed ships capable of fulfilling various roles)
Frigates: A fairly general term (for brevity's sake, covers patrol ships and corvettes) referring to any small, nimble vessel that can serve a variety of purposes. Typically assigned to defensive escort roles to fend off smaller ships, although they can be specialized to serve as torpedo platforms, scout ships, and hit-and-run stealth vessels.
Destroyers: An all-purpose medium-sized vessel that can be used for escort and combat purposes. Normally, they are armed with cannons and missiles, and may be able to carry drones or fighters. Good at hunting down frigates or escaping ships, but can be used as a main combat vessel in fleets composed of small to mid-size vessels.
Cruisers: Long-range, fast-moving ships that are slightly larger than destroyers, cruisers are very combat-oriented ships designed to do immense damage for its size. Missiles make up its arsenal and increase its firepower without the need for having large, complex cannons. Can fill a variety of roles, like the destroyer.
Front-Line Ships (Large ships that make up the bulk of a fleet's combat capability)
Battleships: The main "heavy hitter" class of any fleet, the battleship typically trades maneuverability and speed for heavy armor and heavy firepower. The battleship is the weapon of choice for most offensive and defensive posts, and is the front line of any combat zone. Battleships form the bulk of the large vessels of a fleet and is normally a fleet's main offensive force.
Battlecruisers: A close cousin of the battleship, the battlecruiser combines the firepower of a battleship with the mobility of a cruiser. They are around the same size as a battleship, but sacrifice armor for speed without compromising firepower. A.K.A. "glass cannons".
Carriers: Carriers are very large ships that basically serve as mobile repair and rearming bases for fighters, bombers, drones, and gunships. They can be treated as flagships of many fleets, and have immense sensor and networking capabilities, along with defensive systems that can keep the ship running through a large battle while shooting down swarming spacecraft.
Flagships: Typically the largest and most powerful ship of the fleet, flagships are the main command vessels of a fleet, and can be found leading huge task groups. They are incredibly well-armored, well-armed, and well-guarded, and are often prioritized over other ships; if destroyed it can cripple a fleet's morale and combat capability while boosting the opposing team's confidence.
Super-Capital Ships (Huge, rare vessels capable of taking on entire fleets on their own)
Dreadnoughts: The hugest ship class known to the universe. Very rare and incredibly powerful, typically carries its own fleet and military force with it. Can be equipped with superweapons that can level fleets, planets, or entire dimensions. Verax is the owner of the planet-sized dreadnought The Voyager. Commander Robless owns a fleet of normal-sized dreadnoughts (still well over 300-400km in length, the largest normal flagships only get up to 300km) used as the flagship of his fleets.
Supercarriers: Not quite as massive as the dreadnought but still larger than other front-line vessels, supercarriers serve as the mobile bases of any fleet. Are typically armed with powerful weapons, despite serving a mostly support and command role. They can service just about any ship short of another supercapital.
Civilian Ship Classes
Similar to the ship classes above, but for the non-combat civilian ships that you'll see most commonly.
Transports (Ships designed for carrying people and cargo)
Shuttles: Small, unarmed ships mostly used planet-side to ferry passengers around. Used both as public and private transportation for the citizens of a planet, like we use cars and buses in the real world, and can make it to orbit in order to get people to stations.
Freighters: Large cargo ships that can carry massive amounts of cargo and people alike. Their size makes them relatively durable and have a large cargohold capacity, but are big targets for pirates and raiders, making escorts a must-have.
Private Space Vessel (PSV): Any non-merchant privately owned ship. Yachts, privateer vessels, and most warp-capable civilian ships would be in this category. Usually owned by freelance pilots or larger corporations. Occasionally armed and nimble, capable of holding their own. Most pilots will rent their services out to companies or wealthy civilians, but most stick to patrolled routes in and along trade zones and planets.
Logistics (Specialized spacecraft to fulfill a variety of industrial and logistical needs)
Mining Barges: Large, slow-moving mining platforms designed to mine out asteroids and harvest resources. Oftentimes, you will find groups of these scouring asteroid belts. As with freighters, mining barges are big, unarmed targets, but mining corporations usually clamp down on security, deploying drones and sentry turrets, hiring private security, or sometimes even establishing space stations to protect their assets.
Colony Ships: Large, passenger ships complete with terraforming capabilities and basic contruction materials. Colony ships are sent out to establish life on unoccupied planets or, in extreme cases, are used as long-term evacuation vessels. They travel together in caravans for safety, relying on escorts to protect them from pirates and hostile factions.
Logistics: A general term used for any vessel that constructs, repairs, or supplies. Many of these are non-combat ships, but military fleets also depend on these ships to keep their ships operational in the heat of battle.
"Our deepest fear is not that we are inadequate. Our deepest fear is that we are powerful beyond measure. It is our light, not our darkness that most frightens us." - Marianne Williamson