general relativity - What does it mean to go from a co-variant vector to a contravariant vector?

In most presentations of general-relativity I see the following statement,

We can change from a covariant vector to a contravariant vector by using the metric as follows, ${ A }^{ \mu }={ g }^{ \mu \nu }{ A }_{ \nu }$

My questions are,

1. What is the need to do this particular change in relativity?


2. The covariant components represent the components of a vector the contravariant components represent the components of a dual-vector, for finite dimensional vector spaces the two spaces are isomorphic. What is the significance of representing a quantity in contravariant or convariant forms? Is the need purely mathematical?