Deformations of complex structures and Deformations in quantum algebra I am a bit confused about the meaning of the word "Deformation". 
For one, as in Wikipedia, it seems to refer to fixing a compact surface and varying the complex structure. 
For another, as in Jantzen, Lectures on Quantum groups, there is another definition: Consider the noncommutative associative algebra over a field $k$ generated by $X,Y$ and impose the relation $YX=qXY$ where $q \in k$. This is supposed to be another type of deformation.
Moduli spaces are used in mathematical physics in some way, though I do not know how. This may be just a random coincidence of choice of words; but I wonder if there is any sort of similarity between the two notions of deformation?
 A: The word deformation in these situations have the same meaning. More generally you can deform "any" kind of algebraic and geometric objects such as associative algebras, commutative algebras, lie algebras, modules, complex manifolds, schemes, singularities, vector bundles, coherent sheaves etc
Informally deformation means a family of objects of a given type over some base with distinguished point plus identification of fiber over this point with given object. So you vary point in the base and get different fibers that represents "deformed object".
First important question is what base to choose? In basically determines what kind of deformations you study. If you take base to be just a ring of dual numbers $D=k[t]/(t^2)$ that informally give a tangent direction at a point, you get so called first order deformations, and they correspond to a tangent space of potential moduli space of objects. We can see that this is very important kind of deformations because they give as information about tangent spaces of "moduli space" in particular we know what dimension of moduli space to expect at least at smooth points.
Second choice of base is any commutative artian local ring i.e. $k[t]/(t^n)$, that would correspond to attempt to find higher order "corrections" of first order deformations, it is like solving differential equation by perturbations. On this stage it possible to get obstructions i.e. a get a deformation over $k[t]/(t^n)$, but could not continue as deformation over $k[t]/(t^{n+1})$, such obstructions are cohomology classes in some cohomology theory.
Finally, it is possible to take formal neighborhood of zero $k[[t]]$ as base space that correspond to formal deformation theory, or you can ask questions of convergence and thus consider just small neighborhood of distinguished point, say, in analytic topology.
In your example with algebra you study family over affine line and $q$ is coordinate on this line, point $q=1$ is a distinguished point over this point you get a commutative algebra and as you vary parameter on the base you get family of deformed algebras that are just associative.
It is possible to do exactly the same procedure for complex manifold take a map $X \to \mathbb{C}$, fiber over point $1$ is $X_1$ some given manifold, but fibers over other points could be same complex manifolds that are not isomorphic to $X_1$ but only diffeomorphic i.e.  you indeed vary complex structure and keep underlying smooth manifold the same.
Another possible source of your confusion might be that you see different techniques in your textbooks applied to different situations. For example deformation of complex structure could be described not as some family of complex manifolds over a base, but as deformation of $\bar{\partial}$-operator. This is just another way to look at things via studding differential graded Lie algebra that controls deformations. In complex geometry this so called Kodaira-Spencer algebra, but such differential graded Lie algebra also exists for associative algebras and called Hochschild cohomology complex.
