SQL Server - GROUP BY on one column

You could use a CTE with SUM(Quantity)OVER(PARTITION BY Order_id) + ROW_NUMBER to pick out the desired row from the order-group:

WITH cte 
     AS (SELECT order_id, 
                customer_id, 
                article_id, 
                delivery_date, 
                quantity=Sum(quantity) 
                           OVER( 
                             partition BY order_id), 
                rn = Row_number() 
                       OVER( 
                         partition BY order_id 
                         ORDER BY delivery_date ASC) 
         FROM   orders) 
SELECT order_id, 
       customer_id, 
       article_id, 
       delivery_date, 
       quantity 
FROM   cte 
WHERE  rn = 1 

DEMO

However, your desired result seems to be incorrect (question edited)

This is my result:

ORDER_ID    CUSTOMER_ID ARTICLE_ID  DELIVERY_DATE   QUANTITY
PR10.001    11          20.001a         17-04-2013  3
PR13.001    15          41.022b         19-04-2013  2

Assuming the other columns are functionally dependent on the grouping column, the simplest answers are either

a. Group by the other columns as well:

SELECT Order_id, Customer_id, Article_id, Delivery_date, sum(Quantity)
FROM Orders
GROUP BY Order_id, Customer_id, Article_id, Delivery_date

or

b. Use an aggregate function such as max:

SELECT Order_id, max(Customer_id), max(Article_id), max(Delivery_date), sum(Quantity)
FROM Orders
GROUP BY Order_id

My personal preference is the second approach, as I think it is clearer than the first in indicating which items are actually required for grouping, as opposed to which are merely being grouped to get round the ungrouped/unaggregated column problem.