The correct phrase is "I beg your pardon." You would usually say this when you don't understand what the other person is saying. It's another way of asking someone to repeat what they said so you can hear it again! Sort of like saying "excuse me?"
In the United States, we would say "I beg your pardon" in two situations.
We say it if we don't understand what somebody said.
"The train to Franklin is on time and is going to depart from track [mumble]."
"Pardon?" or, "I beg your pardon?"
We sometimes say it as an intensified way of saying "pardon me." If I'm at the supermarket and I accidentally step into someone's path so that they need to slow down, I say "pardon me." If we actually collide, and it's my fault, and they drop things, then I say "Oh, I beg your pardon, I'm very sorry" (and then I get down on the floor and start picking up the dropped items).