Covariant vs contravariant vectors

We don't talk of covariant and contravariant bases. Start with the basis $\{\mathbf e_i\}$. Then a general vector can be written $$\mathbf v = v^i \mathbf e_i$$ Now if you double the length of a basis vector, you must halve the length of the componant. The components are said to be contravariant, because they change opposite to the basis. In index notation this vector is simply written $v^i$, and we call it a contravariant vector meaning that the components are contravariant.

The inner product

$$ \mathbf u \cdot \mathbf v = g_{ij}u^iv^j $$ prompts the definition $$ u_j = g_{ij}u^i $$ The $u_j$ are components of a vector in the dual space. Because the inner product is invariant, the components $u_j$ change opposite to contravariant components, which means they change in the same way as the basis vectors. They are called covariant components, and we refer to them as covariant vectors.

Technically contravariant vectors are in one vector space, and covariant vectors are in a different space, the dual space. But there is a clear 1-1 correspondence between the space and its dual, and we tend to think of the contravariant and covariant vectors as different descriptions of the same vector.

You have a basis ${\bf e}_i$ in some vector space.

The contravariant components of a vector ${\bf v}$ are given by ${\bf v}=v^i{\bf e_i}$, as Charles Francis says.

The covariant components of a vector ${\bf v}$ are given by $v_i=\mathbf v\cdot\mathbf e_i$

I think that's a more basic way of thinking about them than going in to their transformation properties - though that is of course true.

Incidentally it's then obvious that $\mathbf u\cdot\mathbf v=\sum u_i v^i$ (or $\sum u^iv_i$)

I would say (though mathematicians would disagree and will probably downvote this answer as heretical) that a 'physics' vector is neither covariant nor contravariant. It's a pointing arrow. If you want to do anything useful with it you have to write down its components, which can be either covariant of contravariant.