It is mostly used in England. It is used mostly by the middle and upper class people and not by the working class person. 

How you say" I beg your pardon" can communicate different things. When said gently it can mean "sorry I don't understand" or it implies that you need them to repeat what they said. If you say " I beg your pardon" with more strength and conviction it can imply you have said something that has offended them.

As the previous comments have noted it is a common way to say "I'm sorry" or sometimes people even use it to ask for someone to repeat themselves in American English. It quite literally means 'I ask for your pardon (or forgiveness). Check out this helpful link which gives some great examples: <a href="https://dictionary.cambridge.org/us/dictionary/learner-english/i-beg-your-pardon">https://dictionary.cambridge.org/us/dictionary/learner-english/i-beg-your-pardon</a>;

It is a more formal way of saying sorry. In English US context, it is not that common as it is being used in British English. 

In addition to the above two comments (as an apology or asking to repeat onself), in some situations (in the US) if said in a certain way, it can be used to express outrage or strong disagreement (while appearing civil).

It depends on context.