His name changed throughout his life, and when you're writing about him you should usually use the name he was using in the period you're writing about. At birth his name was Gaius Octavius Thurinus, and generally he was referred to as 'Octavius'. He was adopted by Julius Caesar via his will when he was assassinated in 44 - and in Roman law, when you were adopted you'd take on the names of your adoptive family, and your old nomen (Octavius, in this case) shifted to an -ianus ending (Octavianus) to indicate that it had been your nomen prior to your adoption. So after 44 he was called Octavianus, which has been anglicised to 'Octavian' - except that his political enemies didn't accept the legitimacy of his adoption by JC, and so they refused to call him Octavian, which would imply acceptance that he had been adopted. They continued to call him Octavius.
After he became emperor (or princeps - 'first citizen' - to be more accurate) he took the name Caesar Augustus in 27 BC, Augustus for short, which was used for the rest of his life.
