Leyla, I like the expression 'I beg your pardon', too. I'm both British and old-fashioned, so I do say 'I beg your pardon' sometimes to mean 'I'm sorry', and I also say 'I beg your pardon?' to mean 'Pardon? What did you say?'

We don't use to 'beg someone's pardon' as an everyday verb phrase, though. For example, we don't say 'Jack was late for class so he begged his teacher's pardon'. That really is too old-fashioned. Instead, we'd say 'Jack was late for class so he apologised to his teacher'.

But if you want to say 'I beg your pardon' - as a fixed expression - when you have made a mistake, or if you want someone to repeat something, then that would sound OK.

You are right, Leyla, English people are generally polite. Sometimes we are too polite. We apologise when we have done nothing wrong.

For example, if someone bumps into you in the street or on a bus in England, we BOTH say sorry. The person whose fault it is apologises for bumping into the other person. And the person who has done nothing wrong says sorry for having made the first person say sorry to them.

Sorry isn't the hardest word for the English. In fact it's our favourite word.

Are you asking us whether we can admit when we're wrong?

I think I'm pretty good at it, but I'll never say sorry if I don't believe I've done anything wrong. That isn't to say I harbour grudges - I don't, but I won't apologise to someone just to ease the tension. I think that's what you were asking :D

"I beg your pardon?" is a phrase used by mostly older people. It's just like asking "what did you say?" when you think someone has insulted you (it can be used jokingly as well of course). It's kind of like a test to see whether you'll repeat what you've said. My grandad is 73ish and he says it, and he's English. I doubt you'll here it very often among younger people or outside of England? Maybe it's used in America too, I'm not too sure. It sounds very "English" to me. 

"I beg your pardon?" is used in the US as well, although I agree that it sounds a little British and is used more by older generations. In my opinion it's just a little overly formal for most situations. 

The word you need is 'apologize' ( also spelt  'apologise')  or simply  'say sorry'.