Here is a very rough outline:

Take your family $E$ of elliptic curves over $B := \mathbb{C} - \{0,1\}$. Then take the associated "(co)homology bundle" over $B$, whose fibre over $\lambda$ is the (singular) (co)homology of the elliptic curve $E_\lambda$. To be rigorous, the $i$-th cohomology bundle is $R^i \pi_\ast\mathbb{C}$, where $\pi$ is the map $E \to B$ and $\mathbb{C}$ is the constant sheaf (let us work in the analytic topology). Actually to be precise I should say that $R^i\pi_\ast\mathbb{C}$ is a (locally constant) sheaf of $\mathbb{C}$ vector spaces, and the corresponding vector bundle is $R^i\pi_\ast\mathbb{C} \otimes_\mathbb{C} \mathcal{O}\_B$. It is a fact that these cohomology bundles come with flat (Gauss-Manin) connections $\nabla$. One way to see that the vector bundles are flat is to observe that there are integral lattices $R^i\pi_\ast \mathbb{Z} \subset R^i\pi_\ast\mathbb{C}$.

Let $\omega$ be a 1-form on the family $E$. Note that the 1st cohomology of an elliptic curve is rank 2, so the cohomology bundle $R^1\pi_\ast \mathbb{C}$ is rank 2, thus if we have 3 sections, then they will be (fiber-wise) linearly dependent. So here are 3 sections: $\omega, \nabla_{d/d\lambda}\omega, (\nabla_{d/d\lambda})^2 \omega$. The Picard-Fuchs equation is essentially just the equation which expresses that these sections are linearly dependent. Your equation involving "$\pi$" (not the same as what I am calling "$\pi$") and its derivatives comes from taking this linear dependence equation and "plugging in" (i.e. integrating along) homology classes extended by parallel transport.

The story that I've described above generalizes to arbitrary families of smooth compact varieties.

Thomas Riepe's answer explains some of the more classical reasons why we might be interested in period integrals and Picard-Fuchs equations, so let me say a few words about the relation to Gromov-Witten theory.

The relation to GW theory arises from mirror symmetry, which is a duality between type IIA and type IIB string theories. One of the reasons why mathematicians first became interested in mirror symmetry was because of the prediction of the physicists Candelas-de la Ossa-Green-Parkes in the early 90s that the genus 0 GW invariants of a quintic threefold (type IIA theory) could be computed via an analysis of period integrals and Picard-Fuchs equations coming from a "mirror" family of Calabi-Yau manifolds (type IIB theory). This is a general principle of mirror symmetry: that GW invariants of certain manifolds can be computed via completely different methods on the "mirror manifold". Usually, studying the mirror manifold is "easier" than trying to study the GW theory of the original manifold directly; although by now our knowledge of GW theory has grown considerably, so this is less true than it used to be.

A very nice introductory paper on this material is "Picard-Fuchs equations and mirror maps for hypersurfaces" by David Morrison: http://arxiv.org/abs/alg-geom/9202026

If you're interested in reading further, you should check out the book "Mirror symmetry and algebraic geometry" by Cox-Katz, which covers all of this material in detail and explains the proofs (due to Givental and Lian-Liu-Yau) of the Candelas-et. al. prediction.