To get the components of the contravariant vector $v = v^i e_i$, where $e_i$ is the natural basis, we dot with the basis vectors $e^i$ for the dual space,
$$v\cdot e^j = v^i e_i\cdot e^j = v^i \delta_{i}^j = v^j.$$
Likewise, to find the components of a covariant vector $w = w_i e^i$ we dot with basis vectors from the natural basis,
$$w\cdot e_j = w_i e^i\cdot e_j = w_i \delta^{i}_j = w_j.$$
Sometimes the natural basis vectors are called covariant (since their indices are downstairs) and the dual basis vectors contravariant (since their indices are upstairs).
With this convention a contravariant vector, with contravariant components, is written in terms of the covariant basis!
After a while, you get used to this sort of nonsense.
Addendum: The terms contravariant and covariant refer to how an object transforms under coordinate transformation, $x\to x'$.
In physics, where one is often dealing with coordinates, this is especially vivid.
Does the thing transform contravariantly with $\frac{\partial {x'}^j}{\partial x^i}$ or covariantly with $\frac{\partial {x}^j}{\partial {x'}^i}$?
That is why the terminology is not so bad.
$e^i$ really does transform contravariantly.
This has to be the case so that
$$\begin{eqnarray*}
v &=& v^i e_i \\
&=& v^i \delta_i^j e_j \\
&=& v^i
\frac{\partial {x'}^k}{\partial x^i}
\frac{\partial {x}^j}{\partial {x'}^k} e_j \\
&=& {v'}^i {e'}_i.
\end{eqnarray*}$$
To add another wrinkle, physicists also often say that an object that is invariant under transformation is covariant!
Likewise, the covariant components of v can be obtained from the dot product of v with covariant basis vectors, viz.
