Actually the error I see now is this:
$result = "SELECT password FROM users WHERE username = '$username';" or die(mysql_error());
$password = mysql_query($result);
Should be:
$result = "SELECT password FROM users WHERE username = '$username'";
$password = mysql_query($result) or die(mysql_error());
$password = mysql_fetch_row($password);
$password = $password[0];
Also:
print "Welcome $_SESSION[username]<br>";
Should be:
print "Welcome {$_SESSION['username']}<br>";