" I beg your pardon." I'm not sure how to use this sentence.
does this mean "may you say that again,please?" or "I'm so sorry about that, I hope you could forgive me.."
and I wonder if this sentence could be used in normal conversation or it's too stronger.

Question

Answer

You can use it in both of those situations, to mean either:

I'm sorry I did that.
or
Could you say that again?

NB not 'May you say that again? 'May' is used to ask permission with 'I' or 'we' (eg May I ask a question?)

I don't think it is especially formal. It's an expression, so you are not literally 'begging' the other person, and it's not particularly strong. I use this expression in everyday conversation.

Answer

'I beg your pardon' is a formal and a polite way of saying 'excuse me'.. 
could be used in many different cases..

check the following link for more: http://tinyurl.com/krakglo