[php] $_POST not working. "Notice: Undefined index: username..."

first of all,

be sure that there is a post

if(isset($_POST['username'])) { 
    // check if the username has been set
}

second, and most importantly, sanitize the data, meaning that

$query = "SELECT password FROM users WHERE username='".$_POST['username']."'";

is deadly dangerous, instead use

$query = "SELECT password FROM users WHERE username='".mysql_real_escape_string($_POST['username'])."'";

and please research the subject sql injection