Both correct. Apologise is British Englisn and Apologize is US English.

As you suspected and Saqim has confirmed, the answer was very simple indeed. It's just a regional difference in spelling. But now that you know about one verb, you may as well extend this to many other verbs. Almost any verb ending in this way will follow the same pattern - realise, organise, digitise, materialise and many more - 'ise' in British English and 'ize' in American. You can choose whichever spelling you prefer, though you may find it more convenient to use the 'ize' form. Many users of British English also use the more American 'ize' form, while Americans never use the 'ise' form. Two major exceptions to this rule are 'advise' and 'practise'. British English spells them both with an 's', as you'd expect, while the nouns are spelt 'advice' and 'practice.' American English, confusingly - because it breaks the above rule - spells 'advise' the same as the British way, but spells the verb 'practice' with a 'c', the same as the noun. A (slightly more complex) bonus answer to your simple question.