Regrouping in addition (carrying)
When a column totals ten or more, you can't write a two-digit number in one column. So you 'regroup' ten ones into one ten and carry that ten into the next column. 47 + 8: seven ones plus eight is fifteen , that's one ten and five ones, so you write 5 and carry the 1.
Regrouping in subtraction (borrowing)
When the top digit is smaller than the one below it, you regroup one ten from the next column into ten ones so you have enough to subtract. It's the same trade, just in reverse.
Why place value is the key
Regrouping only makes sense if a child understands that ten ones make one ten and ten tens make one hundred. If carrying and borrowing feel like magic, go back and rebuild place value with base-ten blocks first.