A ‣ Apologies with common replies 
We can apologise [say sorry] in different ways in different situations.
A: I’m (terribly/really ) sorry – I’ve forgotten your book.
B: Never mind. That’s OK.
Terribly/really makes you sound ‘more sorry’.
A: I beg your pardon – I didn’t see you there.
B: That’s all right. (also That’s OK.)
I beg your pardon is a more formal apology, often used if you walk into someone.
A: (I’m) sorry to disturb you.
B: Don’t worry. Come on in. I can finish this later.
We use this phrase when we interrupt / speak to someone who is busy working.
A: (I’m) sorry to keep you waiting – I won’t be long.
B: OK. Fine.
We use this phrase when someone is waiting for us. I won’t be long = I will be with you very soon.
A: Excuse me, I won’t be a minute.
B: OK. Fine.
We use this phrase when we have to leave a room or go somewhere.
A: I must apologise for the noise last night.
B: That’s all right. I understand.
This is a more formal apology, and it is often used in business letters.
A: I’m (really) sorry I’m late.
B: Don’t worry.
Common mistakes
I’m sorry I’m late. (NOT I’m sorry for be late. OR I’m sorry to be late.)
B ‣ Excuses 
We often give an explanation or excuse after an apology. An excuse is a reason for the apology, which may or may not be true. These are excuses students might give for being late for class.
I’m sorry I’m late for class …
– There was a delay / hold-up on the underground. [when a train, plane, etc. leaves or arrives later than you expect]
– I was held up in traffic. [hold up – cause a delay and make someone late – is often used in the passive]
– My train was cancelled [the train company decided not to run the train], and I had to wait half an hour for the next one.
– I overslept [slept longer than I planned or wanted to].
C ‣ Thanks and replies 
These are some common ways of thanking people, with typical replies.
A: Thanks (very much).
B: Not at all. (also That’s OK.)
A: I’ve brought your books.
B: Oh, cheers. (infml)
A: No problem. (infml)
A: I’ll post those letters for you.
B: Oh, thank you. That’s very kind of you. (This is polite and slightly more formal.)