How do you get a ship?

After completing the last class quest (about level 16) on your second planet you will receive your ship.

You will recive a space ship after the following quest:

Planet: Dromund Kaas

• Imperial Agent - Cleaning Up the Mess
• Sith Inquisitor – In The Halls Of Traitors
• Sith Warrior – Leaving Dromund Kaas
• Bounty Hunter – Joy Ride

Planet: Coruscant

• Jedi Consular - Ancient Secrets
• Trooper – Assault On The Works
• Smuggler – Race the Devil
• Jedi Knight – Race to the Ruins

The ship you receive depends on your class:

Jedi Knight & Jedi Consular – Corellian Defender

Smuggler – Corellian XS Stock Light Freighter

Trooper – Rendili Hyperworks BT-7 Thunderclap

Sith Warror & Sith Inquisitor – Fury-Class Imperial Interceptor

Imperial Agent – X-70b-Phantom Prototype

Bounty Hunter – D5-Mantis

Source and official site

You get your ship when your class story arc takes you off the second planet (Dromund Kass or Coruscant). It's not directly related to your level, but I think you should be usually around level 15 at that point.