Note that this answer is outdated! The mysql extension is no longer available out of the box as of PHP7. If you want to use the old mysql functions in PHP7, you will have to compile ext/mysql from PECL. See the other answers for more current solutions.
This would work, see more documentation here : http://php.net/manual/en/function.mysql-fetch-array.php
$result = mysql_query("SELECT names FROM Customers");
$storeArray = Array();
while ($row = mysql_fetch_array($result, MYSQL_ASSOC)) {
$storeArray[] = $row['names'];
}
// now $storeArray will have all the names.