Before anyone moans, I'm not replacing mysql with mysqli/PDO to answer the question. Yes it's wrong that it's used but it's not related to the question.

Correct model: if (there is not an error) { log the person in } else { do something else}.

Your model: check for errors. log the user in anyway.

This is what you're doing now

// checking stuff
if ($_SERVER["REQUEST_METHOD"] == "POST")
{
   if (empty($_POST['username']))
     {$usernameErr = "Username is required.";}
   // blah blah check check check 
}
// don't bother considering the error, just log them in anyway
$sql="SELECT * FROM $tbl_name WHERE usrname='$username'";
// etc

But what you need to do is this:

// check for errors and store them

$errors=array(); // create an empty array to store errors
if (empty($_POST['username'])){
    $errors['usernameErr'] = "Username is required."; // add an error
}else{
    $username =($_POST['username']);
}

if (empty($_POST['password'])){
     $errors['passwordErr'] = "Password is required."; // add an error
}else{
     $password =($_POST['password']);
}

// etc etc

// check if there were any errors anywhere along the way
// and if not, proceed with login
if (!count($errors)) { // check there are no errors
    $sql="SELECT * FROM $tbl_name WHERE usrname='$username'";
    $result=mysql_query($sql);
    $count=mysql_num_rows($result);
    // etc etc
}else{
    // if there were errors do something else
    echo implode("<br />", $errors); // output the errors however you like
}

Try this for a start

<?php

/* validate form first */
if (!empty($_POST['username']))
{ $username = $_POST['username'];
}
else{ echo "Username is required."; }

if (!empty($_POST['password']))
{ $password = $_POST['password'];
}
 else{ echo "password is required."; }


/* Do the queries second i.e */

SELECT * FROM Persons WHERE username='' AND password ='';



?>

hi，You should describe your question clearly，I have read your code and checked it ,when i not fills out password,it was really display Password is required. general validation method is as follows：

if(empty($_POST['username'])){
    $usererror = '...';
    return false;
}else{
    $username = $_POST['username'];
}
if(empty($_POST['password'])){
     $passerror = '...';
     return false;
}else{
   $password = $_POST['password'];
}

The best way to handle error validation is to use same variable, especially if you have many input form data

$username = $_POST['username'];
$password = $_POST['password'];

if ($_SERVER["REQUEST_METHOD"] == "POST") {
    if ($username == '') {
        $error_msg[]= 'Username is required';
    } else if ($password == '') {
        $error_msg[]= 'Password is required';
    }
}

if (!empty($error_msg)) {
    $ERROR_MSG = implode($error_msg);
    exit;
}