The answer depends on precisely which string theory you study, and whether you consider compact or non-compact Calabi-Yaus. Let me focus on the case of compact Calabi-Yaus.

First, let's consider Type IIB string theory on a compact Calabi-Yau threefold $X$.

The "easy" part of the answer is that the vector multiplet moduli space is the moduli space of complex structures on $X$. This is naturally a Kahler manifold of complex dimension $h^{2,1}(X)$, but actually it carries a little more structure, that of a "projective special Kahler" manifold. (In the physics literature you may see this referred to as just "special Kahler".) All of this structure can be read out in terms of the periods of $X$; in particular, it is purely classical geometry, not involving any hoodoo about "quantum corrections".

The harder part of the answer is the hypermultiplet moduli space. Almost the only thing one can say with certainty at the moment is that it is a quaternionic-Kahler manifold (not hyperkahler!) of quaternionic dimension $h^{1,1}(X)+1$. You might wonder, what is it the "moduli space" of? The real answer is that it is "the moduli space of the string theory" but this is not so helpful. To get some grip on things, one could try computing purely classically and ignoring quantum effects. In that approximation, you can identify what the "moduli" in this "moduli space" are: you find that real dimension $h^{1,1}$ is accounted for by deformations of the metric in $X$ (variations of the Kahler class), another $h^{1,1}$ comes from the "$B$ field" (a 2-form on $X$, required to be closed by the equations of motion), another $2 h^{1,1} + 2$ from the "Ramond-Ramond sector" (0,2,4,6-forms on $X$), and finally $2$ from the "axio-dilaton" which I will not try to explain (they are not directly related to the geometry of $X$).

Now, suppose we consider instead Type IIA string theory on a compact Calabi-Yau threefold $Y$.
In this case, things are roughly the reverse of what I wrote above:

First the vector multiplet moduli space.
As in the previous case, this is a projective special Kahler manifold. Its complex dimension is $h^{1,1}(Y)$, and it is sometimes referred to as the moduli space of "complexified Kahler deformations" of $Y$. In some approximation, it has real dimension $h^{1,1}$ coming from deformations of the Kahler metric in $Y$, and another $h^{1,1}$ from a "$B$ field" like the one mentioned above. However, unlike what we had in Type IIB, here the exact special Kahler structure on the vector multiplet space isn't completely determined by the classical geometry of $Y$. Roughly speaking, the additional structure you need is the genus zero Gromov-Witten invariants of $Y$.

(Mirror symmetry is supposed to identify Type IIA on $Y$ with Type IIB on its mirror $X$. In particular, the vector multiplet spaces in those two theories have to be the same. Combining that with what is written above, you would conclude that the periods of $X$ somehow know about the genus zero Gromov-Witten invariants of $Y$. This is indeed the case --- for example, one of the earliest triumphs of mirror symmetry was the computation of the Gromov-Witten invariants of the quintic threefold using the periods of its mirror.)

Next the hypermultiplet moduli space. As in the previous case, it is a quaternionic-Kahler manifold. Its quaternionic dimension is $h^{2,1}(Y) + 1$. In some approximation, you can think of this as decomposed as follows: real dimension $2 h^{2,1}$ from variations of the complex structure of $Y$, another $2 h^{2,1} + 2$ from the "Ramond-Ramond sector" (a 3-form on $Y$), and again $2$ from the mysterious "axio-dilaton".

Note that neither in IIA nor in IIB can you determine the quaternionic-Kahler metric on the hypermultiplet moduli space without dealing with quantum corrections. That makes it a much more challenging object to understand than the vector multiplet moduli space. This is a rather active area of study at the moment, and much more is known now than was known a few years ago, but as far as I know nobody has yet succeeded in constructing a single example.