What is a(n algebro-geometric) family of modular forms? We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to be smooth, proper, etc).
Given such a family, we can take the $\ell$-adic representation associated to any given fiber, and in this sense we also have a "family" of Galois representations. (Alternatively, by the proper base change theorem in étale cohomology, we can take $R^1f_*(\mathbb{Z}_{\ell})$, which is a sheaf on $Y$, and the stalks of this sheaf are the duals of the above Galois representations).
Now consider different question. Can we have a "family" of cuspidal eigenforms whose associated Galois representations fit into a family in the above sense?
I'll consider the case of weight 2 (though I'm most interested in higher weight). Then such family should lead to a family of RM abelian varieties, i.e. those associated to the weight 2 cusp forms.
Let's go back to the elliptic curve (or abelian variety) side a bit, and think about what this would mean. The level of a modular form corresponds to the conductor of the associated elliptic curve, so the level of the modular forms in such a family should be just as bizarre a function of the base as is the conductor of a family of curves.
To try to engineer such a family, suppose we had a family of elliptic curves that were all known to be modular. I'm most interested in rational families, i.e. with open subsets of projective space as bases. Then if the family is defined over $\mathbb{Q}$, we at least know that the fibers of rational points are modular, and we get a "family" of modular forms over the rational points. What would this "family" look like? I have a feeling it would be pretty strange from the point of view of modular forms.
A different approach is to try to to construct a family of modular curves, then view a family of modular forms as a section of the relative cotangent sheaf or some power thereof. Maybe one could try to make it an "eigensection" of some sort of relative Hecke operators. Of course, the very idea of a family of modular curves seems strange, as there are countably-many modular curves!
In fact, this points to a general problem with this attempt: modular forms are based on discrete data, a discrete set of levels, and a discrete set of eigenforms within each level.
I have a feeling that it's impossible to make this notion work, but please let me know if you have good ideas. In particular, it's possible that experts in modular forms and curves would have more ideas.
 A: $\newcommand\Q{\mathbf{Q}}$
$\newcommand\A{\mathbf{A}}$
$\newcommand\AQ{\A_{\Q}}$
How about instead of considering lisse sheaves, derived pushforwards, étale cohomology, modular forms, blah blah blah, you consider instead the following situation: over the affine line $\Q[t]$, you can consider the equation $x^2 - t$; it's a family of quadratic extensions. If you like, you can turn this into a smooth map of curves (eliminating the bad fibre at $0$) $\pi: X \rightarrow Y$, and you can consider the lisse sheaf $R^0 \pi_* \mathbf{Z}_l$, the stalks of which are Galois representations  corresponding to the quadratic character of the associated quadratic extension (along with the trivial character, which I'll suppress). All the Galois representations corresponding to rational points are automorphic/modular for $\mathrm{GL}_1(\AQ)$ - that's a theorem of Gauss called quadratic reciprocity. What does this family look like? Well, it is what it is; the global data is related to the factorization of $t$ in the expected way, and it's not really a "family" is any analytic sense. There is, however, one useful fact to observe about how this family behaves as one varies $t$. Namely, the characters $\chi_t$ are locally constant. That is, if $t$ and $s$ are close in $\Q_v$ for any place $v$, then the local characters $\chi_{t,v}$ and $\chi_{s,v}$ are equal. For example, if $t$ and $s$ are both the same sign, then $\chi_{t,v}(-1) = \chi_{s,v}(-1)$. This is Krasner's lemma (we use smoothness here). It turns out that this "local constancy" of Galois representations is true more generally, I think Kisin proved something along these lines (although you should think of that result as also being Krasner's Lemma).   
A: Perhaps it is easier to explain what's going on here in the context of Galois representations. I know of two almost wholly disjoint ideas that are, confusingly, both described using phrases like "families of Galois representations". 


*

*One can consider "families of representations" of any group, which are just homomorphisms from G to GL_n(A) for some (usually commutative) ring A, or more generally GL(V) for some locally free sheaf V on a base scheme S, etc. These are "families" in the sense that the image of a group element is a matrix whose entries are functions on Spec(A) (resp. on S, etc). Note that, intuitively, the group is fixed and the coefficients are varying.

*One can also consider the kind of "geometric" family you mentioned in your question and Michigan J Frog enlarged upon: given a family of geometric objects over a base S, you can do various kinds of relative cohomology to give sheaves on S whose fibres have an action of some kind of Galois group depending on the fibre, and in particular the generic fibre has an action of something like the fundamental group of S. So here the group is, so to speak, varying in the family as well.
The first kind of family, over a p-adic base and with G being a Galois group, comes up a lot in the context of modular forms (Hida, Coleman-Mazur, etc). These can be viewed as sections of a family of sheaves on a subvariety of the rigid-analytic space you get by analytifying the modular curve. Note that we are varying the coefficients and not the group, again. 
The second kind of family doesn't come up so much in modular form theory, although it makes a notable appearance in Kato's work on Iwasawa theory for modular forms.
