You can use PlayerPrefs to save level unlocks.

e.g.

PlayerPrefs.SetInt("Level2Unlocked",1);

To check if a level is unlocked you go like so:

if(PlayerPrefs.HasKey("Level2Unlocked")){
  Application.LoadLevel(2);
}

Check out the docs for other data types you can save in player prefs

You would use Application.LoadLevel or Application.LoadLevelAdditive or Application.LoadLevelAsync or Application.LoadLevelAdditiveAsync to load scenes. See Application for more details.

To preserve data between scenes, you would want to use DontDestroyOnLoad or like spinalJack said, use player prefs.

Please see the latter portion of the 3D Platformer Tutorial for example usage.