how can i store this array into a session and use sessions to move the elements inside the array up/down/left/right diagonally
$board = array(A B C D E F G H
Call session_start and afterwards store your variables in $_SESSION -- they will be available throughout the session:
session_start();
$_SESSION['board'] = array( ... );
Moving elements is just a matter of assigning one value to another, for example:
$_SESSION['board'][$new_i][$new_j] = $_SESSION['board'][$old_i][$old_j];
$_SESSION['board'][$old_i][$old_j] = ...;