Why is it that mirror symmetry has many relations with algebraic geometry, rather than with complex geometry or differential geometry? (In other words, how is it that solutions to polynomials become relevant, given that these do not appear in the physics which motivates mirror symmetry?) I would especially appreciate nontechnical answers.
Here are a few scattered observations:
Our ability to construct examples (e.g. of CY manifolds) is limited, and the tools of algebraic geometry are perfectly suited to doing so (as has been noted).
Toric varieties are a source of many examples -- Batyrev-Borisov pairs -- and they are even "more" than algebraic, they're combinatorial. In fact, the whole business is really about integers in the end, so combinatorics reigns supreme.
The fuzziness of $A_\infty$ structures is more suited to algebraic topology rather than geometry.
Continuity of certain structures (which are created from counting problems) across walls, scores some points for analysis over combinatorics and algebra.
Elliptic curves are only "kinda" algebraic, and the mirror phenomenon there is certainly transcendental.
Physics indeed does not care too much about how the spaces are constructed, but (as has been noted) even the non-topological version of mirror symmetry is an equivalence of a very algebraic structure (which includes representations of superconformal algebras).
I was hoping to unify these idle thoughts into a coherent response, but I don't think I can. Maybe the algebraic geometric aspects just grew faster because the mathematics is "easier" (or at least better understood by more mathematicians): witness the slow uptake of BCOV and its antiholomorphicity within mathematics.
To respond personally: these days, I try to transfer the algebraic and symplectic structures to combinatorics so that I can hold them in my hand and try to understand them better.
Here is my impression ...
(I am very much a non-expert in the physics (and probably the mathematics too) so I may well be wrong about some of these things.)
Algebraic geometry sometimes enters the picture in string theory and physics because, while we start with, say, a compact Kähler manifold, for some reason or another we maybe get an integral Kähler class (for example see this MO question), and thus our manifold is projective by the Kodaira embedding theorem, and thus it is algebraic by Chow's theorem. Conversely, we may be actually interested in possibly non-algebraic compact Kähler manifolds in the physics or string theory, but the algebraic manifolds will provide at least a pretty big class of nice examples to play with.
And at least for smooth projective algebraic varieties, GAGA theorems tell us that many things (like for example, sheaf cohomology) are the same whether we consider our space as an algebraic variety or as an analytic thing. For the B-model side of mirror symmetry, I think this is how algebraic geometry (as opposed to complex analytic geometry) generally comes into play --- via GAGA theorems or at least "GAGA principles". For example, it is a fact that analytic coherent sheaves on smooth projective varieties are algebraic. From this it follows that, at least for smooth projective varieties, the derived category of coherent sheaves is the same whether we look at things algebraically or analytically. (I'm guessing, but I don't know for a fact, that in the physics the analytic objects are the a priori relevant ones.)
Another interesting issue is the fact that algebraic geometry often appears even on the A-model side of mirror symmetry, which is supposed to be the symplectic side of the story. I don't really know anything about this, so maybe someone else can say more, but there's some work on, for example, the relation between the symplectic version of Gromov-Witten theory and the algebraic geometry version of Gromov-Witten theory -- they're supposed to coincide in the case of smooth projective varieties. It's perhaps not too surprising, since the symplectic version of GW theory involves J-holomorphic curves after all, but it's definitely not a trivial result.
I suppose the naive explanation for the appearance of surfaces is that they're worldsheets of strings, but I don't really know the explanation for why the surfaces should have complex structures, i.e. why they should be Riemann surfaces (and by the way, it is also a basic fact that any compact Riemann surface is algebraic) nor do I know why the maps from the curves to the target manifolds should be holomorphic or J-holomorphic. I hope that other MO users, especially people who know about string theory and physics, can say more about these things...
Kevin Lin gave a great technical answer to this question (which should be accepted) but I would like to add some more "philosophical" reasons for this:
 algebraic geometry methods are easier to apply and much more well-developed. I don't mean algebraic geometry is easy, I just mean that the tools, by their nature, give more concrete results (for example, on toric varieties), as opposed to geometric analysis methods, which by their nature often yield non-constructive or non-explicit results.
 the algebraic geometers got into the Mirror Symmetry game much earlier and made more rapid progress than the differential geometers. (And they wrote many of the books.)
I would still appreciate more answers, though. Perhaps people who are very adept at having a dual existence (like Eric Zaslow) can contribute their opinions?
Part of the physics motivation for mirror symmetry involves properties of the chiral ring of N=2 superconformal field theories. Some of these have a description in terms of the polynomials appearing in algebraic geometry. One of the earliest references on this is Algebraic Geometry and Effective Lagrangians, Emil J. Martinec, Phys.Lett.B217:431,1989. There are many papers discussing the relation between these "Landau-Ginzburg" models and mirror symmetry. See for example the paper by Berglund and Katz, http://arXiv.org/pdf/hep-th/9406008.
The following is a rough outline of the most elementary structures that appear in a physics discussion of mirror symmetry. It turns out that the physics actually leads in two ways directly to polynomial equations that describe the varieties. On the most basic level string theory deals with Calabi-Yau manifolds that provide the extra dimensions needed to go from 10 dimensions to the physical 4 dimensions that we live in. Calabi-Yau manifolds are described by polynomials, hence it is not too unexpected (in retrospect) that the pheonomenon of mirror symmetry would be discovered by constructing enough polynomials describing enough Calabi-Yau spaces. And so it was. On a more fundamental, string theoretic level, the conformal field theory on the string worldsheet has a mean field theory limit in which is described by a so-called Landau-Ginzburg potential. This Landau-Ginzburg potential in turn has a classical limit in which it describes a polynomial that defines a hypersurface in a toric variety. It is precisely this polynomial that describes the Calabi-Yau variety corresponding to the underlying conformal field theory. Mirror symmetry is a simple operation on the worldsheet, defining a sign flip in the charge of the fields, but it is not too surprising that this operation on the worldsheet is reflected in the form of the polynomial, hence the precise structure of the Calabi-Yau space.
It's probably slightly offbeat, verging on the mystical, and my apologies if it sounds a bit ridiculous, but I reckon mirror symmetry may ultimately derive from sets of degrees of freedom $x_i$ satisfying:
$ x_1 + x_2 .. + x_N = 0 $
$ x_1 x_2 .. x_N = 1 $
For small N, such as N = 4, this variety is birationally equivalent to all kinds of different forms, some with a tantalizingly "physical" appearance.
Also, for larger N, it can clearly "split" (either exactly or approximately) into the union of lower-dimensional varieties of the same form.
Obvious symmetries are $ x_i \rightarrow 1 / x_i $ and (for even N) $ x_i \rightarrow - x_i $, and I dare say there are others.
It would be very interesting to know if these varieties are Calibi-Yau manifolds. But that would be better discussed in another thread.