Dubrovin's conjecture (or Bayer's modified version, if prefer) establishes a condition for the semisimplicity of the quantum cohomology of a manifold X, but, Why is important to know that the quantum cohomology of some manifold X is semisimple?
[Math] Semisimple quantum cohomology
ag.algebraic-geometry
Related Solutions
The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomorphic curves in symplectic topology"). PSS is a canonical ring isomorphism from $QH^{\ast}(X)$ to the Hamiltonian Floer cohomology of $X$, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal.
Now to Hochschild cohomology of the Fukaya category $F(X)$. There's a geometrically-defined map $QH^{\ast}(X) \to HH^{\ast}(F(X))$, due to Seidel in a slightly different setting (see his "Fukaya categories and deformations"), inspired by the slightly vague but prescient remarks of Kontsevich from 1994. One could define this map without too much trouble, say, for monotone manifolds. It's constructed via moduli spaces of pseudo-holomorphic polygons subject to Lagrangian boundary conditions, with an incidence condition of an interior marked point with chosen cycles in $X$. The question is whether this is an isomorphism.
This statement is open, and will probably not be proven true in the near future, for a simple reason: $QH^*(X)$ is non-trivial, while we have no general construction of Floer-theoretically essential Lagrangians.
There are two positive things I can say. One is that Kontsevich's heuristics, which involve interpreting $HH^{\ast}$ as deformations of the identity functor, now have a natural setting in the quilted Floer theory of Mau-Wehrheim-Woodward (in progress). This says that the Fukaya category $F(X\times X)$ naturally embeds into the $A_\infty$-category of $A_\infty$-endofunctors of $F(X)$.
The other is that for Weinstein manifolds (a class of exact symplectic manifolds with contact type boundary), there seems to be an analogous map from the symplectic cohomology $SH^{\ast}(X)$ (a version of Hamiltonian Floer cohomology on the conical completion of $X$) to $HH^{\ast}$ of the wrapped Fukaya category, which involves non-compact Lagrangians. (Edit August 2010: I was careless about homology versus cohomology. I should have said that $HH_{\ast}$ maps to $SH^{\ast}$.) Proving that this is an isomorphism is more feasible because one may be able to prove that Weinstein manifolds admit Lefschetz fibrations. The Lefschetz thimbles are then objects in the wrapped Fukaya category.
One might then proceed as follows. The thimbles for a Lefschetz fibration should generate the triangulated envelope of the wrapped category (maybe I should split-close here; not sure) - this would be an enhancement of results from Seidel's book. Consequently, one should be able to compute $HH_{\ast}$ just in terms of $HH_{\ast}$ for the full subcategory generated by the thimbles. The latter should be related to $SH^{\ast}$ by ideas closely related to those in Seidel's paper "Symplectic homology as Hochschild homology".
What could be simpler?
ADDED: Kevin asks for evidence for or against $QH^{\ast}\to HH^{\ast}$ being an isomorphism. I don't know any evidence contra. Verifying it for a given $X$ would presumably go in two steps: (i) identify generators for the (triangulated envelope of) $F(X)$, and (ii) show that the map from $QH^{\ast}$ to $HH^{\ast}$ for the full subcategory that they generate is an isomorphism. There's been lots of progress on (i), less on (ii), though the case of toric Fanos has been studied by Fukaya-Oh-Ohta-Ono, and in this case mirror symmetry makes predictions for (i) which I expect will soon be proved. In simply connected disc-cotangent bundles, the zero-section generates, and both $HH_{\ast}$ for the compact Fukaya category and $SH^{\ast}$ are isomorphic to loop-space homology, but I don't think it's known that the resulting isomorphism is Seidel's.
Added August 2010: Abouzaid (1001.4593) has made major progress in this area.
Actually, the semisimplicity should hold with no hypotheses on X, so no example should exist. In fact it is generally expected that, with char. 0 coefficients and over a finite field (both hypotheses being necessary), every mixed motive is a direct sum of pure motives -- so the question for arbitrary varieties reduces to that for smooth projective ones.
The reason is as follows: the different weight-pieces have no frobenius eigenvalues in common (by the Weil conjectures), so the weight filtration can be split by a simple matter of linear algebra. (And the splitting will even be motivic since frobenius is a map of varieties.)
Edit: In response to Jim's comment, let me try to provide a clearer argument (2nd edit: no longer using the Tate conjecture). I claim that if we assume the existence of a motivic t-structure over F_q w.r.t. the l-adic realization in the sense of Beilinson's article http://arxiv.org/pdf/1006.1116v2.pdf, then provided that H^i_c(X-bar) is Frobenius-semisimple for smooth projective X, it is in fact so for aribtrary X.
Indeed, given a motivic t-structure, its heart is an artinian abelian category where every irreducible object is a summand of a Tate-twist of an H^i(X) for X smooth an projective, and furthermore there are no extensions between such irreducibles of the same weight (this is all in Beilinson's article).
That's all true over a general field. But now let's argue that, in the case of a finite field, there also can't be extensions between such irreducibles of different weights; then in the motivic category all of our H^i_c(X-bar) of interest will be direct sums of summands of H^i(X)'s, and we'll have successfully made the reduction to the smooth projective case.
So suppose M and N are irreducible motives of distinct weights over F_q, and say E is an extension of M by N. Consider the characteristic polynomials p_M and p_N of Frobenius acting on the l-adic cohomologies of M and N. By Deligne, they have rational coefficients and distinct eigenvalues, so we can solve q * p_N == 1 (mod p_M) for a rational-coefficient polynomial q. But then (q*p_N)(frobenius) acting on E splits the extension (recall from Beilinson's article that the l-adic realization is faithful under our hypothesis), and we're done.
Later commentary: apparently, when I wrote this I was a little too excited about the perspectives offered by motives. I should emphasize the point essentially made by Minhyong Kim, that the reduction from the general case to the proper smooth case likely doesn't require any motivic technology, and should even be independent of any conjectures. One just needs to know that there's a weight filtration on l-adic cohomology of the standard type where the pure pieces are direct sums of direct summands of appropriate cohomology of smooth projective varieties. As Minhyong says, this probably follows from Deligne's original pure --> mixed argument, via use of compactifications and de Jong alterations. Or at least that's what it seems to me without having gone into the details. I'm sure someone else knows better.
Best Answer
One reason is Givental's conjecture, which says that in the semisimple case, genus 0 GW invariants determine higher genus GW invariants. See this paper of Teleman, in which the conjecture is proved.
The theory of Frobenius manifolds in general is quite complicated. I guess semisimple Frobenius manifolds form a relatively tractable set of examples. Here are some basic references for the theory of semisimple Frobenius manifolds:
Manin: Frobenius manifolds, quantum cohomology, and moduli spaces
Dubrovin: Geometry of 2d topological field theories
Lee, Pandharipande: Frobenius manifolds, Gromov-Witten theory, and Virasoro constraints (so far unpublished and incomplete; available at Pandharipande's webpage)
Semisimple Frobenius manifolds also arise in singularity theory, when studying for instance isolated hypersurface singularities (see Hertling's book Frobenius manifolds and moduli spaces for singularities; the three references above probably also talk about this), or in the physics terminology "Landau-Ginzburg (B-)models". I don't know whether Frobenius manifolds (in particular non-semisimple ones) arise more generally in singularity theory...? In any case, these Frobenius manifolds coming from singularity theory are supposed to be related to those coming from Gromov-Witten theory via mirror symmetry.*
Another comment: Quantum cohomology of, for example, $\mathbb{P}^n$ is semisimple. Then perhaps this makes quantum cohomology and GW theory of projective varieties more tractable, because of quantum Lefschetz ... but I don't really know anything about this. But very roughly speaking, I think this is the strategy of Givental in his proof of the "mirror conjecture" of Candelas et. al. regarding the genus 0 GW theory of the quintic 3-fold, though I might be wrong.
*Edit: For example, this paper of Etienne Mann seems to prove a mirror theorem relating the quantum cohomology Frobenius manifolds of (weighted) projective spaces and the Frobenius manifolds associated to the mirror Landau-Ginzburg B-models. As Arend mentions, germs of semisimple Frobenius manifolds are specified by a finite set of data, and I think the strategy of Mann's paper is to show that these data coincide for the two Frobenius manifolds.