"x equals x" is written with 2 times an "=", not one time ;). You are now setting a new value for $n in each if() statement. You should write it as $n == 3 instead of $n = 3.
Also you're not comparing your results correctly. In line 2 of the script you present, you set $n to be a result set, not the actual result. The actual result would be $n['alias_switch'].
Try this code, if you wish:
<?php
$result = mysql_query("SELECT alias_switch FROM peoplebase");
$n = mysql_fetch_assoc($result);
$alias_switch = $n['alias_switch'];
if($alias_switch == "1") {
mysql_query("UPDATE peoplebase SET alias_switch='2'");
}
elseif($alias_switch == "2") {
mysql_query("UPDATE peoplebase SET alias_switch='3'");
}
elseif($alias_switch == "3") {
mysql_query("UPDATE peoplebase SET alias_switch='1'");
}