This would be a quick macro to do so:
Private Sub Worksheet_SelectionChange(ByVal Target As Range)
'note, this won't copy to row 1, leaving headers intact.
Target.EntireRow.Copy (.Rows(.Rows.Count + 1).EntireRow)
However, using it with selection change, can be quite annoying, because it will be executed every time you select another cell - so you might want to think about using some sort of backchecking or another way to execute this.
Maybe just use the code as a custom macro and assign a shortkey.
When you want to do this only with formulas, you'll have to setup each cell of your destination row with a formula. I know of no way to change a cell directly by using a formula, except of course the cell with the formula.
My suggestion then would be, to use eigther named-ranges, VLOOKUP or MATCH.
A named-range might get quite close to what you stated:
MYRANGE is defined as
This would then give you row 14 of the destination sheet. This will also work, if A1 is on the SourceSheet , just edit it accordingly.
Just as easily you could use two named-ranges:
By changing MYSOURCERANGE to another row, you would then effectivly change the source of MYDESTRANGE, while using the same row as before.