One must distinguish between quantum/classical on the string world-sheet and in spacetime.
Both of your statements are basically correct, but should read something like "CFT theory is the space of classical solutions to the spacetime equations of string theory" and "Quantization of the
the world-sheet sigma model of a string theory gives rise to a CFT."
In a little more detail,
the sigma-model describing string theory propagation on some manifold M is a 2-dimensional
quantum field theory which in order to describe a consistent string theory must be a conformal
field theory. The "classical limit" of this 2-dimensional field theory is a limit in which some
measure of the curvature of M is small in units of the string tension. To construct a CFT one
must solve the sigma-model exactly, including world-sheet quantum effects.
The coupling constants of the sigma-model are fields in spacetime such as the metric $g_{\mu \nu}(X(\sigma))$ on $M$ where $X: \Sigma \rightarrow M$ define the embedding of the string world-sheet $\Sigma$ into $M$. Now there
is also a spacetime theory of these fields. You can think of it as a ``string field theory". At low-energies it can sometimes be usefully approximated by a theory of gravity coupled to some finite number
of quantum fields, but in full generality it is a theory of an infinite number of quantum fields. Roughly speaking, each operator in the CFT gives rise to a field in spacetime. The spacetime string field theory lives
in 10 dimensions for the superstring or 26 dimensions for the bosonic string and it also has a classical limit. The classical limit is $g_s \rightarrow 0$ where $g_s$ is
a dimensionless coupling constant. It appears in perturbative string theory as a factor which
weights the contribution of a Riemann surface by the Euler number of the surface. It can also be
thought of as the constant (in spacetime) mode of a scalar spacetime field known as the dilaton.
The main point is that there are two notions of classical/quantum in string theory, one involving
the world-sheet theory, the other the spacetime theory. In order to avoid confusion one must be clear which is being discussed. Unfortunately string theorists often assume it is clear from the context.
In response to the further question about the space of string fields, I would suggest that you have a look at the introductory material in http://arXiv.org/pdf/hep-th/9305026. You may also find http://arXiv.org/pdf/hep-th/0509129 useful. I should add that while string field theory has had some success recently in the description of D-brane states, it is not widely thought to be a completely satisfactory definition of non-perturbative string theory.
My opinion is that physicists transferred from study of "individual objects" to that of "large systems" where the order arises from limit probability laws rather than from simple deterministic formulae and from the study of something "readily observable" to something that is, essentially, "a purely mathematical object" invisible to a direct experiment. This brought them to the realm traditionally reserved for pure mathematicians. And, of course, with their eagerness to use whatever tools they have available in any way that is short of total lunacy, they went on to make predictions, many of which could be confirmed experimentally, leaving a long trail of successes and failures in their wake for mathematicians to explain.
I do not know the situation with the string theory and low dimensional topology but I have some idea about what's going on in random matrices (thanks to Mark Rudelson and his brilliant series of lectures) and in percolation/random zeroes (thanks to Stas Smirnov and Misha Sodin). The thing that saves physicists from making crude mistakes there is various "universality laws".
Here is a typical physicist's argument (Bogomolny and Schmidt). You want to study the nodal domains of a random Gaussian wave $F$ (the Fourier transform of the white noise on the unit sphere times the surface measure). Let's say, we are in dimension 2 and want just to know the typical number of nodal lines (components of the set $\{F=0\}$) per unit area. The stationary random function $F$ has only a power decay of correlations. However,
we ignore that and model it with a square lattice that has the same length per unit area as $F$ (this is a computable quantity if you use some standard integral geometry tricks). Now, at each intersection of lattice lines, we choose one of the two natural ways to separate them (think of the intersection as of a saddle point with the crossing lines being the level lines at the saddle level). Then, we get a question (still unresolved on the mathematical level, by the way) about a pure percolation type model. Thinking by analogy once more, we get a numerical prediction.
From the viewpoint of a mathematician, this all is patented gibberish. There is no way to reduce one process to another (or, at least, no one has the slightest idea how this could be done as of the moment of this writing). Still, the Nature is kind enough to make the answers the same or about the same for all such processes and Mathematics is powerful enough to provide an answer (or a part of an answer) for some models, so the physicists run a simulation, and, voila, everything is as they predicted and we are left with 20 years or so worth of work to figure out what is really going on there.
I'm not complaining here, quite the opposite: this story is really quite exciting and the work mentioned is both real and fascinating. We are essentially back to the days when Newton tried to explain the nature of gravity looking at Kepler's laws trying various options and separating what works from what doesn't. I'm only saying that the famous "physicists' intuition", which is so overrated, is actually just the benevolence of Nature. Why should the Nature be so benevolent to us remains a mystery and I know neither a physicist, nor a mathematician, who could shed any light on that. The best explanation so far is contained in Einstein's words "God is subtle, but not malicious", or, in a slightly less enigmatic form, "Nature conceals her mystery by means of her essential grandeur, not by her cunning".
Best Answer
In addition to Chris Gerig's operator-language approach, let me also show how this magical number appears in the path integral approach.
Let $\Sigma$ be a compact surface (worldsheet) and $M$ a Riemannian manifold (spacetime). The string partition function looks like $$Z_{string}=\int_{g\in Met(\Sigma)}dg\int_{\sigma\in Map(\Sigma,M)}d\sigma\exp(iS(g,\sigma)).$$ Here $Met(\Sigma)$ is the space of Riemannian metrics on $\Sigma$ and $S(g,\sigma)$ is the standard $\sigma$-model action $S(g,\sigma)=\int_{\Sigma} dvol_\Sigma \langle d\sigma,d\sigma\rangle$. In particular, $S$ is quadratic in $\sigma$, so the second integral $Z_{matter}$ does not pose any difficulty and one can write it in terms of the determinant of the Laplace operator on $\Sigma$. Note that the determinant of the Laplace operator is a section of the determinant line bundle $L_{det}\rightarrow Met(\Sigma)$. The measure $dg$ is a 'section' of the bundle of top forms $L_g\rightarrow Met(\Sigma)$. Both line bundles carry natural connections.
However, the space $Met(\Sigma)$ is enormous: for example, it has a free action by the group of rescalings $Weyl(\Sigma)$ ($g\mapsto \phi g$ for $\phi\in Weyl(\Sigma)$ a positive function). It also carries an action of the diffeomorphism group. The quotient $\mathcal{M}$ of $Met(\Sigma)$ by the action of both groups is finite-dimensional, it is the moduli space of conformal (or complex) structures, so you would like to rewrite $Z_{string}$ as an integral over $\mathcal{M}$.
Everything in sight is diffeomorphism-invariant, so the only question is how does the integrand change under $Weyl(\Sigma)$. To descend the integral from $Met(\Sigma)$ to $Met(\Sigma)/Weyl(\Sigma)$ you need to trivialize the bundle $L_{det}\otimes L_g$ along the orbits of $Weyl(\Sigma)$. This is where the critical dimension comes in: the curvature of the natural connection on $L_{det}\otimes L_g$ (local anomaly) vanishes precisely when $d=26$. After that one also needs to check that the connection is actually flat along the orbits, so that you can indeed trivialize it.
Two references for this approach are D'Hoker's lectures on string theory in "Quantum Fields and Strings" and Freed's "Determinants, Torsion, and Strings".