Hi julianthemath,
There are a number of other ways of getting the result you did, but here a couple of easy ones:-
Method 1:
Enter the following into the cells as shown.
A1: 1
A2: =A1^2
Left-click in A1 to select it, and while still holding down the mouse button move the cursor into B1 to select both of them together.
Release the mouse button, but don't click anywhere on the page, otherwise those two cells will be deselected.
Left-click the 'fill-handle' that now appears in the bottom right-hand corner of B1 (it's a black square, which changes to a cross-hair when you hover the mouse over it), and while holding down the left mouse button drag it all the way down column 2 to row 25.
That automatically fills columns 1 and 2 in one go. 
Method 2:
Enter the following into the cells as shown.
A1: 1
A2: =A1^2
Left-click in A1 to select it.
While holding down the keyboard's Ctrl key, left-click the fill-handle (explained in Method 1) and drag it down column 1 to row 25.
The sequence of numbers 1:25 now appears down the 25 rows in column 1.
Left-click in B1 to select it, and double-left-click the fill-handle to automatically fill column 2 with the formula from B1 for every cell that has data to the left of each column 2 cell.
There...done! 
And don't let anonimnystefy fool you...I'm no Excel expert! I've learnt a few things over the years by trying this and that, mainly for puzzles and work. Excel has a lot going for it, including the enormous amount of information and help available on the internet.