OK, you followed my example a bit too closely. The text you see in green that's preceded by an apostrophe isn't actually code. The apostrophe forces VBA to see it as a comment. So what's happening is that you're not actually copying anything, therefore not pasting anything as a result. Here's what it should be:
If Range("A1") = 1 Then
Range("D3").Copy Range("B3")
Else
Range("D6").Copy Range("B6")
End If
Note that if you see "Select" followed by "Selection" you can generally eliminate both statements, which will really help your code. That's what I did, and why the code looks so short.
Now, since you're only copying single cell ranges, you can eliminate copying altogether:
If Range("A1") = 1 Then
Range("B3") = Range("D3")
Else
Range("B6") = Range("D6")
End If