How to get the client printer name using JavaScript code

You need a source code and I need to see some effort made by you before asking so!

As it has already been answered, you wont be able to know the Client system details until you use some ActiveX kind of control (but that would be only IE specific).

Any website/page hosted has access on the server machine and not on client system.
Though, based on complete scenario you might have some other option. Like if you want to print stuff on client printer then you can directly use the print feature of browser from javascript to do that for you.

<script language="javascript" type="text/javascript">

        var filename;
function winPrintDialogBox(filename) {
    var oShell = new ActiveXObject("WScript.Shell");
    sRegVal = 'HKEY_CURRENT_USER\\Software\\Microsoft\\Windows NT\\CurrentVersion\\Windows\\Device';

    var sName = oShell.RegRead(sRegVal)
    if (sName == '') {
        alert('Please, Check the Default Printer');
    }
    document.getElementById("hdnResultValue").value = sName;
    return sName;
}
function HandleError()
{
    alert("\nNothing was printed. \n\nIf you do want to print this page, then\nclick on the printer icon in the toolbar above.")
    return false;
}
    </script>