The Preserve Column/Row checkboxes put $'s in the proper places in the cell/range reference. Or you can type them into the references yourself.
The differences in how the references are treated:
A1 : This address will be relative to the cell with the formula. Say that A1 is 3 columns to the left and 2 rows up from the formula. If you copy/paste the formula to a cell in a different column and row, A1 will adjust to be 3 columns to the left and 2 rows up from that cell.
A$1: This preserves the row. Given the same example, the reference will adjust to be 3 columns to the left of the formula but will stay in row 1.
$A1: This preserves the column. Given the same example, the reference will adjust to be 2 rows up from the formula but will still be in column A.
$A$1: This preserves both. When you copy/paste the formula to another cell, the reference will still be $A$1.
The exact same things apply to range references except now you have a starting cell and an ending cell and you may have to apply settings to both so it will do what you want it to do. The settings can be, and often are, different for each.