Many two- and three-digit addition problems do not require regrouping. In the problem 45 + 32, for example, you can easily add the ones column, 5 ones plus 2 ones equal 7 ones. And you can easily add the 10s column, four 10s plus three 10s equal seven 10s. Your answer is 77 and you didn't need to regroup.
Let's say you're adding 42 + 39. Now, when you add two ones plus nine ones you get 11 ones. You cannot just write down 11 because you'll have a problem when you try to add the tens column and you already have a one in the 10s column for an answer. Broken down, 11 is the same as one 10 and one one. So when you add 2 + 9, you write a 1 in the ones column, and regroup the other 1, which represents one 10, to the 10s column. Then you add 4 + 3 + the 1 you moved to the 10s column when you regrouped, and you have eight 10s. Your answer is 81.
Many times you can subtract without regrouping. If the number you're subtracting (the number on the bottom) is smaller than the number you're subtracting from (the number on the top), you don't need to regroup. For example, in the problem 75 - 43, you can easily subtract the ones column, because three is smaller than five. So, 5 - 3 = 2. You can also easily subtract the 10s column because four is smaller than seven. So 7 - 4 = 3. Your answer is 32, and you didn't need to regroup.
In the example 36 - 19, when you try to subtract the ones, you realize that nine (the number on the bottom) is larger than six (the number on the top). You can regroup. The three in the 10s column means you have three groups of 10 ones. So you can move one group of 10 to the ones column, giving you a group of 16 ones. Now you can subtract the ones: 16 ones minus 9 ones equals 7 ones. In the 10s column, you regrouped and moved one group of 10, you were left with 2 groups of 10. So you subtract two 10s minus one 10 equals one 10. Your answer is 17.