Did you post the complete code? If so, you've got some serious problems here:
<?php
if ($_POST['action'] == 'delete')
{
$sql = "DELETE FROM azizpassword WHERE user_id = {$_POST['user_id']}";
$conn = mysql_connect("localhost","root","") or die("Could Not Connect To The Database");
mysql_select_db("aziz",$conn) or die("Could Not Select The Database");
You define the correct query in $sql to delete the user, but... you're not even connected to the database yet! Once you connect, you don't even run the $sql query; you go on to select the users again!
You need to reverse the order and execute the $sql statement like so:
<?php
if ($_POST['action'] == 'delete')
{
$conn = mysql_connect("localhost","root","") or die("Could Not Connect To The Database");
mysql_select_db("aziz",$conn) or die("Could Not Select The Database");
$sql = "DELETE FROM azizpassword WHERE user_id = {$_POST['user_id']}";
mysql_query($sql) or die('Error running query: ' . mysql_error());
If that solves your problem, you might also want to search the board for solutions to 'SQL Injections'. I've made several posts, all including the same code (using [man]get_magic_quotes_gpc/man to determine if magic_quotes_gpc is On or not, and then using [man]addslashes/man or [man]mysql_real_escape_string/man appropriately).