Inner joins will be your friend:
SELECT * FROM mydb INNER JOIN anotherdb ON anotherdb.product id = mydb.product id
The inner join sets up a relationship between the two tables. With this query, it will find all relationships. You could use the where clause to narrow the query down. You may find a left join handy as well (maybe you have a product but it doesn't have a full name - the inner join will not show that product but a left join would - depends on how your data is setup and how you are using it).