Apologize and apologise both mean to say sorry or express regret for a mistake, and neither is more "correct" than the other. Apologize is standard in American English and in Oxford's British style, while apologise is the spelling most British writers reach for, as in "He apologised to his sister for forgetting her birthday."
The core difference
The only difference is spelling convention — the meaning, to say sorry, is identical in both.
- apologize — say sorry for a mistake or wrongdoing: The company apologized publicly after the data breach.
- apologise — the same word in everyday British spelling: He apologised to his sister for forgetting her birthday.
Which spelling should you use?
Apologize comes from apology plus the verb-forming suffix -ize, which itself goes back to the Greek apologia, "a speech in one's defense." That -ize form is the one American English has kept, and it is also the spelling Oxford dictionaries favour even for British text.
Apologise simply swaps in the -ise ending that most British writers use across this whole family of verbs, alongside organise and recognise. Pick apologize for American readers or Oxford house style, and apologise for general British writing.