Here is a very rough answer.
The Gromov-Witten invariants show up in a few a priori different
contexts within string theory. Let me focus on one particular place they show up that is
directly related to conventional physics, as opposed to topological
quantum field theory.
Type IIA string theory is formulated on a spacetime "background"
which is, in the simplest setup, just a Lorentzian 10-manifold. The
equations of motion of the theory require (at least in their leading
approximation) that the metric on this 10-manifold should be Ricci-flat.
A popular thing to do is to take this 10-manifold of the form
X x R^{3,1}, where X is a compact Calabi-Yau threefold.
We can simplify matters by taking X to be very small ---
smaller than the Compton wavelength of any of the particles we are able
to create. (Remember that in quantum mechanics particles have a
wavelike character, with wavelength inversely related to their energy;
since we only have limited energy available to us, we can't make
particles with arbitrarily short wavelength.) A little more precisely,
let's take X such that the first nonzero eigenvalue of the Laplacian is
larger than the energy scale we can access.
In this case we low-energy
observers will not be able to detect X directly in any experiments. To
us, spacetime will appear to be R^{3,1}.
What will be the physics we see on this R^{3,1}? We will see
various different species of particle. Each species of particle that
we see corresponds to some zero-mode of the Laplacian of X.
In particular, there are particles corresponding to classes in H^{1,1}(X).
The genus 0 Gromov-Witten invariants are giving
information about the interactions between these particles. (So if you want to calculate what will come out when you
shoot two of these particles at each other, one of the inputs to that calculation
would be the genus 0 Gromov-Witten invariants.) The higher genus Gromov-Witten
invariants are giving information about interactions which involve these particles
together with other particles related to the gravitational interaction.
Donaldson-Thomas invariants in mathematics are a virtual count of sheaves (or possibly objects in the derived category of sheaves) on a Calabi-Yau threefold. In physics, sheaves (and more generally objects in the derived category) are considered as models for D-branes in the topological B-model and Donaldson-Thomas invariants are counts of the BPS states of various D-branes systems. For example, the "classical" DT invariants that are considered by MNOP count ideal sheaves of subschemes supported on curves and points. You will hear physicists refer to such invariants as "counting the states of a system with D0 and D2 branes bound to a single D6 brane". The single D6 brane here is the structure sheaf $\mathcal{O}_X$ and the D0 and D2 branes form the structure sheaf $\mathcal{O}_C$ of the subscheme $C$ (which is supported on curves and points) and the term "bound to" refers to the map $\mathcal{O}_X \to \mathcal{O}_C$ because they are replacing the ideal sheaf with the above two-term complex (which are equivalent in the derived category. Note that the $k$ in D$k$-brane refers to the (real) dimension of the support.
There is a discussion of the meaning of the motivic DT invariants in physics in the paper "Refined, Motivic, and Quantum" by Dimofte and Gukov (http://arxiv.org/pdf/0904.1420) where the basic claim is that the motivic invariants and the "refined" BPS state counts are the same. "Refined" here refers to the way you count BPS states. BPS states are certain kinds of representations of the super-Poincare algebra and "counting" means just finding the dimension of these representations (I think that little book on super-symmetry by Dan Freed has a good mathematical discussion of this). Sitting inside the super-Poincare algebra is a copy of $\mathfrak{sl}_2\oplus\mathfrak{sl}_2$ and normally one looks at the action of the diagonal $\mathfrak{sl}_2$ on the space of BPS representations and finds the dimensions of the irreducibles, for the "refined" count, you look at both copies of $\mathfrak{sl}_2$. The generating function for the dimensions of these representations thus gets an extra variable which is suppose to correspond to the Lefschetz motive $\mathbb{L}$ in the motivic invariants.
As for the DT/GW correspondence, I'm afraid that I don't really understand the physicist's explanations. There is a few paragraphs in MNOP (presumably written by Nekrasov) about it and I think that physicists regard it as well understood, but I haven't found something that I can understand. Let me know if you do.
Best Answer
Nima Arkani-Hamed had a series of talks at JHU roughly 6 months ago which I attended related to this topic. He discussed it at Stony Brook a little bit over a week ago (pointed out by Emilio Pisanty in the comments above in which he used the term "amplituhedron", but my understanding of this comes mostly from his earlier talks.
Update: I managed to track Nima down today and get him to explain the details. Surprisingly almost everything was correct, but the loop level description has some changes. I've also made some aesthetic changes, attempting to follow the notation in Trnka's slides as closely as possible while being readable by mathematicians. Also note that Nima claimed that their paper on this would appear "very soon".
Update 2: Their paper has appeared on the arXiv: http://arxiv.org/abs/1312.2007. At first glance everything seems consistent with what I've written below. I think it should be readable to mathematicians simply by skipping the few sections which require knowledge of physics. They do find some combinatorial results therein which may be of interest, but as this answer is already rather large I'll just direct those interested to the paper directly.
Everything here is over $\mathbb R$. We know that the quotient of the subset $\tilde{M}(k,n) \subset M(k,n)$ of matrices with full rank by the $GL(k)$ left action gives the Grassmannian $G(k,n)$, in which each matrix is mapped to its row space. Define $M_+ (k,n)$ to be the subset of $M(k,n)$ in which all $k \times k$ minors are positive (ordering the rows in the process). The image of $M_+(k,n)$ under this quotient is the positive Grassmannian $G_+(k,n)$.
Fix a matrix $Z \in M_+(k+m,n)$ as input data, which comes from physics, but won't really affect the combinatorial structure* at all.
Then there is a map $Y_{n,k,m}: G_+(k,n) \rightarrow G(k,k+m)$ given by $([C],Z) \mapsto [CZ^T]$, where brackets denote equivalence classes under the $GL(k)$ action (the requirement that $CZ^T$ has full rank is automatically satisfied if both are in their respective positive pieces). Its image $\mathcal P_{n,k,m}$ is called the tree-level amplituhedron. The combinatorial structure does depend on $n,k,$ and $m$. This case is apparently fairly well understood thanks to their work with Alexander Postnikov (according to Nima).
(One of the interesting aspects of this positive-real story is that while the map $Y_{n,k,m}$ is only a rational map of algebraic varieties, its base locus doesn't intersect $G_+(k,n)$.)
*I've seen it claimed that $Z$ can be taken to live in $G_+(k+m,n)$. I must admit this doesn't make sense to me because the map described above is not invariant under the right action by $GL(k+m)$. If I'm missing something obvious though than feel free to correct me. At the very least, I'm pretty sure that the above description works, but it might be a bit more redundant than necessary.
As I alluded to above, $\mathcal P_{n,k,m}$ is not the full amplituhedron. Rather, it's just the tree-level case. The full amplituhedron has another nonnegative integer parameter $l$ which determines the loop order. This gives additional coordinates to points in the amplituhedron. The subregion of $M(k+2l, k+m)$ of interest which the $C$ vary through satisfies somewhat more stringent positivity constraints than those of the ordinary positive Grassmannian that we had above in the case $l=0$. I will call this $l$-positivity, though this is my own terminology. $C' = \left( \begin{matrix} C \\ C^{(1)} \\ \vdots \\ C^{(l)} \end{matrix} \right)$ with $C$ as $k \times n$ and each $C^{(i)}$ as $2 \times n$ is $l$-positive iff for any $I = \{i_1 , \ldots, i_r\} \subseteq \{1, \ldots, l\}$ (with $i_1 < i_2 < \cdots < i_r$), the submatrix $\left( \begin{matrix} C \\ C^{(i_1)}\\ \vdots \\ C^{(i_r)} \end{matrix} \right) \in M_+(k+2r,n)$ (including the case $I=\phi$), and each $C^{(i)}$ is only well-defined up to addition of elements of $C$.
For convenience of notation let $\mathcal A_{n,m} = Y_{n,2,m}$. The $l$-loop amplituhedron is then the image of $\{[C'] | C'\text{ is }l\text{-positive}\} \times \{Z\}$ in $G(k,k+m) \times (G(2,k+m))^l$ by just applying $Y_{n,k,m}$ to $([C],Z)$ and $\mathcal A_{n,m}$ to each $([C^{(i)}],Z)$ (all with the same $Z$). This is called $\mathcal P_{n,k,l,m}$ (Trnka drops the $m$, presumably since $m=4$ for physics). The space that this is embedded in has no significance either combinatorially or physically. We could take it to be in the $l$-fold product of the $G(2,m)$ bundle over $G(k,k+m)$ such that the fiber at each point is those $2$-planes orthogonal to that $k$-plane.
It is important to realize that the amplituhedron itself is not so much the object of interest: rather, that is a meromorphic volume form defined on it (and on the Grassmannian, or bundle over Grassmannian, in which the amplituhedron is Zariski-dense). The principal job of the amplituhedron is to help nail down this form: the form is required to be well-defined on the interior of the amplituhedron. It seems very hard to make this statement mean anything without talking about positive real parts of varieties.
The parameters are important for physics, so I'll list them, but of course if you're not interested in the physics you can set them to be whatever you like. $k$ is the order in perturbation theory. $l$, the loop order. $m=4$ is the case for physics, but in principle $m$ can be any even positive integer ($m=2$ makes the loop part trivial, so $m=4$ is in some sense the first interesting case). $n$ is the number of momenta in the scattering process. $Z$ is a positive matrix that represents all of the momenta, but at least for the purpose of combinatorics the structure doesn't really depend on the choice of $Z$. There are of course cases in which the construction does not make sense; these are irrelevant for physics (e.g. $n-k < m$ is unphysical). Also, while I'm talking about physics, to get physical predictions out of the amplituhedron, there is a particular volume form which is simply integrated over the region. This volume gives the amplitude for the process.