Products are definitions, they are way that it makes sense to take 'x' of 'y'. With complex numbers this is what I am assuming you mean by "simple product", however the same definition doesn't make sense with vectors. The dot product is just one way of defining a product between two vectors, if we assume they are composed of 3 components the dot product is defined as...
[tex] < a_1, a_2, a_3 > \cdot < b_1, b_2, b_3 > = a_1b_1 + a_2b_2 + a_3b_3 [/tex]
An alternative way of defining vector products is the "cross product"
I will point out (may it be known) that the cross product of two vectors is a vector where as the dot product of two vectors is a scalar. Two completely different things.