From the wikipedia articles on coproducts and products:
"Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects."
"The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. (...) coproducts can be and typically are dramatically different from products."
I can see that in the coproduct explanation, each object does not need to have a morphism to some object, whereas the product does need that.
I am just trying to find an example that makes this difference (or any "dramatic difference") clearer. I guess the overall question would be, why does the dual notion provide such a different and a lot of times richer structure if "all we are doing" is reversing the source and target of morphisms and their order of composition?