I
noun
BAD : I beg your pardon, but I was very busy and couldn't spare the time.
GOOD : I'm sorry, but I was very busy and couldn't spare the time.
BAD : I'd like to beg your pardon because I was late.
GOOD : I'm sorry I was late.
GOOD : Please forgive me for being late.
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I beg your pardon is used (1) to apologize to a stranger because you have bumped into them by accident, sat in their seat by mistake, etc: 'I beg your pardon. I didn't know the tabel was reserved.'
(2) to politely ask someone to repeat what they have said: 'Does this bus go to Marble Arch?' 'I beg your pardon?' 'This bus, does it go to Marble Arch?'
(3) when someone has said something that makes you feel surprised, shocked, angry etc: 'Who's that man with the long nose?' 'I beg your pardon! That happens to be my husband.'
Note that in formal situations you can use excuse/forgive me instead of I'm sorry : 'Please excuse me for taking so long to answer your letter.'
II
verb
BAD : I asked the teacher if I could be pardoned for a few minutes.
GOOD : I asked the teacher if I could be excused for a few minutes.
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pardon (formal) = forgive: 'I'm sure they will pardon the occasional mistake.'
excuse = give someone permission to stay away from school, work etc, or leave a classroom, meeting etc: 'Can I be excused from swimming today please? I've got a cold.'