When you apologize to someone, you say that you are sorry that you have hurt them or caused trouble for them. You can say 'I apologize' as a formal way of saying sorry.
双语例句
例：
I apologize for being late but I have just had a message from the hospital.
我为迟到表示歉意，但我刚得到了一个来自医院的消息。
例：
He apologized to the people who had been affected.