To just count the different items for one customer you can use
SELECT COUNT(DISTINCT order_name)
FROM orders
WHERE client_id=1;
If you want a summary count for every customer you can use a group by
expression:
SELECT client_id, COUNT(DISTINCT order_name)
FROM orders
GROUP BY client_id;
If you want the latest row for every customer and order_name, you can use a subquery:
SELECT id, client_id, order_name
FROM
(
SELECT max(id) AS id
FROM orders
GROUP BY client_id, order_name
) single_orders
INNER JOIN orders ON single_orders.id = orders.id
ORDER BY client_id, order_name;
This would select the maximum id for each distinct customer-order-combination and then get the corresponding row data.
Finally, in MySQL you can also get a comma-separated list of the distinct items ordered by any customer:
SELECT client_id, group_concat(DISTINCT order_name ORDER BY order_name)
FROM orders
GROUP BY client_id
ORDER BY client_id;