I apologize for the delay.对于延迟，我非常抱歉。
I apologize...和I'm sorry...都可在致歉时使用，但前者的语气更加郑重。 因某亊延迟而致歉的时候,可以说I apologize to you for the delay.当道歉 的对象很明确时，就可以省略to you,直接说I apologize for the delay.就 可以了。
We finally shipped your order today. I apologize for the delay. We included a small gift as a token of apology. Please accept it and we hope you'll enjoy it.
您订购的货物我们已于今天发出。对于延迟，我非常抱歉。为 表达歉意，我们随货物给您寄去了一份小礼物，希望您能够接 受并喜欢。

I hope you understand.希望您能谅解。

在双方关系亲近的情况下，可以省略主语，直接说Hope you understand. 像“Ihopeyou+动词”的句型，可以表达“拜托”的语气。
I couldn't send you an e-mail sooner to let you know about the change of plans. I hope you understand.
未能早些发邮件吿知您方案的变更，希望您能谅解。
Sorry about the last-minute change. Hope you understand.
很抱歉，发生了临时变更，希望您能谅解。