If we number the ships shown in the hanger of the Advanced Edition of the game by position:
1 2 3 4 5
6 7 8 9 10
This is briefly how to get the ships.
Expect minor spoilers throughout
The default ship, always available.
Reach sector five with the Kestrel Cruiser.
In a Rebel Stronghold sector you may locate a "huge Rebel shipyard".
Choose to look around, this will trigger combat with a difficult opponent.
Defeat this opponent.
Complete the game (defeat the Rebel Flagship) with the Engi Cruiser.
In a Zoltan Homeworld sector you may be approached by a ship claiming to be an unarmed peace envoy.
Choose the Hear them out type option and you'll get a quest marker.
Once there you have to negotiate peacefully - first choose the reconcile option, then the bloodshed is bad option.
Complete the game (defeat the Rebel Flagship) with the Federation Cruiser.
Unlock 4 ships, not including the starting Kestrel
Complete a quest involving stolen blueprints, that you can find in an Engi Homeworld sector.
You need to have an Engi crew member.
One node will have civilian ships frantically discussing something, choose the blue option with your Engi crew member.
You receive two quest markers - one real, one false, with no way to tell apart - go to either (but plan for reaching both) and fight the ships, when they try to surrender demand info on the stolen tech.
If you defeated the one at the real quest marker a new final marker will appear.
Head to the final quest marker and defeat the ship you find there.
Complete the game (defeat the Rebel Flagship) with the Rock Cruiser.
Complete a quest that you can find in an Rock Homeworld sector.
One node will involve a Rock ship insulting you.
Respond with the option along the lines of "I'll succeed or die trying".
Follow the quest marker wait for the ship to FTL away. Don't destroy them, and don't kill all their crew. (You can cause them some damage, and you can kill some crew, if you need to.)
Go to the final quest marker.
Complete the game (defeat the Rebel Flagship) with the Slug Cruiser.
Complete a quest that you can find in an Slug Home Nebula sector.
Requires level 2 sensors or a Slug crew member.
Slug ships will often surrender and offer their stores to be spared, accept all of these as one will offer an experimental weapon, refuse this and ask for data instead.
At the new quest node choose "Try to tail them without being noticed," After that, if you have level 2 scanners or a slug, choose the blue option, then destroy the enemy ship or its crew before it powers up its FTL drive.
Complete the game (defeat the Rebel Flagship) with the Mantis Cruiser.
Complete a quest involving a famous thief. This can be found in a Mantis Homeworld sector.
You need to have a Mantis crew member as well as a teleporter system and level 2 medbay or clonebay.
There will be a node where a captain with the name KazaaakplethKilik will greet you, use the 'blue' response that is available if you have a Mantis crew member.
Kill all the crew on the ship - do not destroy it.
Then choose the blue option again (requires level 2 teleporter).
Finally choose another blue option (requiring the level 2 medbay).
Complete the game (defeat the Rebel Flagship) with the Zoltan Cruiser.
This ship is found by first acquiring the 'Damaged Stasis Pod' augmentation - as a reward for a random encounter found in Pirate Sectors, Engi Sectors/Homeworlds and Rock Sectors/Homeworlds.
With this installed, you have to wait until you find a Zoltan research facility (found in Engi Sectors/Homeworlds and Zoltan Sectors/Homeworlds), in addition to the normal options is a blue option to have them examine the capsule.
You will then aquire a new, special, crew member.
Now you need to find the Rock Homeworld sector. Here you may find a random event involving an 'ancient device', using your specialist crew member you have to trigger the blue response option.
You will suddenly travel to a new, hidden, sector. In this sector is a quest marker. Go to this quest marker and this new ship will unlock, the Crystal Cruiser.
Complete the game (defeat the Rebel Flagship) with both Type A and Type B of every ship except the Lanius Cruiser.
###Type B (All Ships)
Complete two of the three ship achievements for the same type of ship.
###Type C (All Ships except Lanius and Crystal, that have A and B only)
Reach sector 8 with Type B of the same ship and Advanced Edition Content enabled.
I've parsed information from here and here.