The two different versions of MacBook Air have different hard drive connectors. Failure to perform the action appropriate to your machine will damage your hard drive! Be very careful when disconnecting the hard drive cable. The ZIF connectors are fragile.

If you have the original MacBook Air (A1237), use a spudger or your thumbs to flip up the ribbon cable retaining flap TOWARD the cable (seen in picture 1).

If you have the A1304 version of MacBook Air, use a spudger or your thumbs to flip up the ribbon cable retaining flap AWAY from the cable (seen in picture 2).

Pull the hard drive cable straight away from its socket.