What does it mean for a differential equation "to be integrable"? What does it mean for a differential equation "to be integrable"?
Are "integrable" and "solvable" synonyms?
The first thing that comes to my mind is to say: it's integrable if we can find the analytic solution (a function that can be written as y = f(x1, x2, ...)) that when plugged into the differential equation satisfies it.
Sometimes we cannot find such a function in closed form, so we use numerical methods to solve the differential equation. Is the equation still defined "integrable" in this case? If yes, when a differential equation is not integrable?
In the case of the Nonlinear Schrodinger equation, which is a nonlinear partial differential equation:


*

*we say that the equation is "integrable" because we can solve it using the inverse scattering method, but also this method can provide closed form solutions only in particular cases.

*we can also solve it using finite difference numerical methods. Because of this could we have said that the equation was "integrable" even before discovering the inverse scattering method?


EDIT:
Since apparently the definition of "integrability" is not straigthforward, my main concern is to understand if the inverse scattering technique to solve the equations like the Nonlinear Schrodinger equation or the KdV equation is equivalent to a Finite Difference technique to say that we "solved" the euqation. 
Moreover, "solving a differential equation" is equivalent to say "solve the initial value problem"? I guess that when numerical methods are applied (so that we cannot get an analytic solution), we are actually solving the IVP, is that correct?
 A: Integrability has a specific meaning for certain differential equations arising in geometry, but I'm not sure if it has a broader meaning for more general differential equations.
Given a collection of $1$-forms $\omega_i$ on a manifold $M$, a submanifold $N \subseteq M$ is said to be integral if the tangent space of $N$ lies in the kernel of each $\omega_i$ at every point.  $N$ is maximally integral if the tangent space of $N$ is precisely the intersection of the kernels of the $\omega_i$'s at every point.  And the collection of $\omega_i$'s is said to be integrable if $M$ admits a foliation by maximally integral submanifolds.
A typical example of an integrable system according to these definitions is the $1$ from $dx$ on $\mathbb{R}^2$: the kernel of $dx$ at each point is spanned by the tangent vector $\partial y$, and the span of $\partial y$ at a point $p$ is precisely the tangent space of the vertical line passing through $p$.  Since the set of all vertical lines forms a foliation of $\mathbb{R}^2$, we're done.
Interesting examples of non-integrable systems come from contact geometry.
So at least in this context integrability means "solutions exist and have nice geometric structure".  The actual definition that I gave makes sense only for first order systems, but probably this is the right intuition for higher order systems.  In particular, I don't think how the solutions are actually found is important for this terminology.
A: There is also an old-fashion meaning   attached to the attribute  integrable,    namely solvable  by quadratures, where  quadrature is  old-fashion speak for  integrals, definite or indefinite.  An  equation was also considered integrable, if you could  describe its solutions as convergent power series. So  an equation was considered integrable if you could solve it using a combination of these two methods. An implicit  description of solutions was considered acceptable.    
In the 19th century, long before  the advent  of computers, the search   for integrable  equations was a very honorable     endeavor.   Even a celebrity such as  H. Poincare  started his  mathematical career trying to produce  large classes of integrable differential equations. This is in fact the content of his  dissertation. This  search  for integrable equations lead him to ask one very natural question. 
Taking for granted that all linear differential systems are integrable (using for example the Jordan decomposition of the matrix of the system) is it possible to reduce any (real analytic) differential system to a linear one, via a clever choice of local coordinates?
Away from stationary points this is always possible so the question boils down  to understanding what happens  near a stationary point. In his dissertation   Poincare solves a special case of this problem (linearization near  stable  stationary point). He  could not  solve the general case  since he was stymied by the so called small denominators. Carl Siegel  eventually bypassed this problem in the 1930s, more that 50 years after  Poincare's dissertation.   
The problem    of small denominators     haunted the mathematicians since.  In the 1960s the Russian mathematician Brjuno  (some other spellings of this name are used)  described  very refined sufficient conditions allowing one to bypass  the small-denominator conundrum. At the end of last century Yoccoz showed that Brjuno's conditions are   also necessary.  (He even received a Fields medal for this and other work.) 
I have   to mention two other   remarkable    contributions around the concept of  integrability. The first  is  the Kolmogorov-Arnold-Moser theorem which refers to the special concept of integrability described by  Paul Siegel in his answer.  The old fashion concept of integrability is present in  Hovanskii's concept of pfaffian functions, concept  that has a very good model theoretic  incarnation (it leads to a  very useful $o$-minimal category).
