What is the correct spelling advertise or advertize?

Either spelling is correct. The spelling "advertise" is the British spelling whereas the spelling "advertize" is the American spelling of the Word. The same is true for Words like organise/organize, recognise/recognize, or realise/realize. The former in these cases is British, the latter American. Some confusion may occur as often a spell check system will accept only one of these and is not always consistent as to which it preferences.