Lots of click handlers that hide one element and show another
Extract the duplicated code to a function:

function showOneHideOther(elToShow, elToHide) {
    $(elToShow).show();
    $(elToHide).hide();
}


You can reuse the function anyplace now:

$('#Back').on('click', function() {
    showOneHideOther('#TableRow', '#FormRow')
});
$('#AddLanguageShow').on('click', function() {
    showOneHideOther('#FormRow', '#TableRow')
});

$('table').on('click', 'button.edit', function (evt) {
    showOneHideOther('#FormRow', '#TableRow')
    languageManager.load(getRowId($(this)));
});