You issue here is that you have told the database how these two tables are 'connected', or what they should be connected by:
Have a look at this image:

~IMAGE SOURCE
This effectively allows you to 'join' two tables together, and use a query between them.
so you might want to use something like:
SELECT COUNT(B.order_id), A.cust_id, A.first_name, A.last_name
FROM customer A
LEFT JOIN cust_order B //this is using a left join, but an inner may be appropriate also
ON (A.cust_id= B.Cust_id) //what links them together
GROUP BY A.cust_id; // the group by clause
As per your comment requesting some further info:
Left Join (right joins are almost identical, only the other way around):
The SQL LEFT JOIN returns all rows from the left table, even if there are no matches in the right table. This means that if the ON clause matches 0 (zero) records in right table, the join will still return a row in the result, but with NULL in each column from right table. ~Tutorials Point.
This means that a left join returns all the values from the left table, plus matched values from the right table or NULL in case of no matching join predicate.
LEFT joins will be used in the cases where you wish to retrieve all the data from the table in the left hand side, and only data from the right that match.
Execution Time
While the accepted answer in this case may work well in small datasets, it may however become 'heavy' in larger databases. This is because it was not actually designed for this type of operation.
This was the purpose of Joins to be introduced.
Much work in database-systems has aimed at efficient implementation of joins, because relational systems commonly call for joins, yet face difficulties in optimising their efficient execution. The problem arises because inner joins operate both commutatively and associatively. ~Wikipedia
In practice, this means that the user merely supplies the list of tables for joining and the join conditions to use, and the database system has the task of determining the most efficient way to perform the operation. A query optimizer determines how to execute a query containing joins. So, by allowing the dbms to choose the way your data is queried, you can save a lot of time.
Other Joins/Summary
- AN INNER JOIN will return data from both tables where the keys in each table match
- A LEFT JOIN or RIGHT JOIN will return all the rows from one table and matching data from the other table.
- Use a join when you want to query multiple tables.
- Joins are much faster than other ways of querying
>=2 tables (speed can be seen much better on larger datasets).