I have 3 tables that I want to combine, see below for details:
You need two joins:
SELECT
product.productID,
category.categoryID,
product.name,
product.price,
category.name
FROM product
JOIN product_cat ON product.productID = product_cat.productID
JOIN category ON category.categoryID = product_cat.categoryID
If a product could be in no categories and you still want to return it, change JOIN to LEFT JOIN in both places.
An alternative approach:
SELECT
product.productID,
product.name,
product.price,
GROUP_CONCAT(category.name)
FROM product
JOIN product_cat ON product.productID = product_cat.productID
JOIN category ON category.categoryID = product_cat.categoryID
GROUP BY product.productID
However it might be better just to use two queries instead of putting multiple values into a single cell.
You can use group_concat if using mySQL...
see this thread.
SQL to join one table to another table multiple times? (Mapping products to categories)
(possible duplicate)