Relation between componenet and algebraic definition of covariant vectors

Question

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its components by orthogonal projection we achieve covariant components. For latter case we need reciprocal basis in order we can expand the vector by its covariant components. But I read somewhere, in fact covariant vectors are elements of dual space and they expanded by dual basis. My question is : how is the latter approach to covectors equivalent to previous approach ? Furthermore, I need some good explanation for latter aproach to covariant vectors. Thank you.