Ships that can be built in-game are:
Fighters, Engineers, Frigates, Colony ships Destroyers, Bombers, Heavy Cruisers, Landers and Battleships.
Also: Unmanned Platforms mod 1 and 2
Fighters, Frigates and Destroyers are the best early to mid game warships that can be built, Fighters are your sponges they are weak alone but strong in numbers (Do not be-afraid to destroy some in order to free up pilots), Frigates are your scouts (they have a tech that can detect ships hidden by an undercover base) and haulers, (they have the biggest hold size and are the fastest ships in the game) Destroyers are what make up the bulk of your fleet they are good at killing Defensive platforms and other light ships.
Engineers and Colony Ships these are some of the weakest ships that can be built, Engineers are used to harvest debris after battles as well as harvest resources from comets. This unit is also required to construct a Space Base there must be at least 600 of them present until construction has been completed (please note having more than 600 does not speed up the construction time), if they are destroyed you will have a small amount of time to replace them before the station self-destructs. (you will also need 3 Colony ships these will be consumed upon starting)
Colony ship tech is an early game tech that will allow you to expand, you must have the right amount of them when selecting a planet and your culture level will determined whether you can colonize another planet or system. These ships are expensive to build like the Landers and Bombers tho not as costly, they move incredibly slow and are easy targets, you should keep them hidden by your undercover Base or an allied one. (please note that if you attempt to retreat them or another fleet within the system that's under attack they will become visible during the hyperspace countdown it may be possible to catch an enemy fleet that arrives or leaves a system a few seconds after you arrive e.g. Engineers on a comet harvest.
As you progress you will unlocked better ships like the Bomber, Heavy Cruiser, Lander and the Battleship tech, all of these require a mountain of resources to field, only the most prosperous empires will meet the requirements and be-able to build thousands of them. These ships can only be built at the Large Shipyard and at certain upgrade levels.
Bombers and Landers are some the most expensive units in the game Landers being more costly then bombers. Bombers can be acquired just after mid game these ships should be well protected and split apart from the war fleet if you send them together the warships will not shoot at the defenders since the objective of the bombers is set to bombing the other ships will go by those orders as well they will simply act as a meat shield until your fleet is destroyed or the bombing time elapsed, same can be said for the Landers except they will be set to invade rather than bomb. Landers are a late game tech that can be used to take over Space Bases and planets of enemy players (Ancient planets are the only planets you cannot take or bomb) they not only are expensive but they require a huge amount of pilots to build. (please note in order to take a system that is just under 30000 population you must build 300 Landers,1 Lander =100 population, in-game it says 300 population however this is false) Also these ships are extremely slow.
Heavy Cruisers and Battleships these units are the most powerful ships you can field. Heavy Cruisers are a late game tech that will counter medium class ships, a few thousand of these will make short work of a frigate and destroyer fleet. (please note if your enemy is fielding an extreme amount of destroyers such as 77k ships, Cruisers will not counter those odds but will eat a chunk out of them) They are very expensive and require the Large Shipyard to produce. Most players skip this tech and go straight on to Battleship tech.
Battleships are the best most powerful ship that can be built, capable of both engaging fleets in space and bombing Space Bases/planets (please note the damage dealt when bombing is 1/4 of a bomber's planetary attack) these warships are very expensive and slow to produce, they require a large amount of pilots they are the last ship tech to unlock and most players will never get that far. They will kill anything in their path but remember to support them with Destroyers because Defensive Platforms excel at killing heavy classed ships although defenders will need hundreds of thousands of them to make use of their advantage.
Unmanned Defensive Platforms these are not ships rather static sentries of a system, once built they cannot move between systems they will simply guard your planets against attacks but remember that if your undercover base is on they will not defend unless the ship count including platform count exceeds the the camouflage limit they will become viable and defend or if the enemy arrives with a frigate fleet bigger than the camouflage limit they will un-cloak you. Later you can research Unmanned Platform Mod 2 an upgrade to it's predecessor.
You can also acquire an Ancient Cruiser by successfully taking down a level 8 anomaly, however please note that this ship cannot be repaired so be warned.
After starting a new game 48 hours later an event will trigger a story mission this will tell you to attack a level 2 anomaly when won will reward you with 4 Colony ships.
Ships that cannot be owned by a player includes:
Ancient warships (except the Cruiser of the ancients) and the Unknown warships: Light Spaceship, Corvette, Bomber and Heavy Cruiser.
Written by Covenant please add anything if I've missed something.