Simply put, you need to rewrite all of your database connections and queries.
You are using mysql_* functions which are now deprecated and will be removed from PHP in the future. So you need to start using MySQLi or PDO instead, just as the error notice warned you.
A basic example of using PDO (without error handling):
<?php
$db = new PDO('mysql:host=localhost;dbname=testdb;charset=utf8', 'username', 'password');
$result = $db->exec("INSERT INTO table(firstname, lastname) VAULES('John', 'Doe')");
$insertId = $db->lastInsertId();
?>
A basic example of using MySQLi (without error handling):
$db = new mysqli($DBServer, $DBUser, $DBPass, $DBName);
$result = $db->query("INSERT INTO table(firstname, lastname) VAULES('John', 'Doe')");
Here's a handy little PDO tutorial to get you started. There are plenty of others, and ones about the PDO alternative, MySQLi.