As Kevin comments, Hochschild homology and cohomology are defined for any $A_\infty$-category $\mathcal{A}$. That includes Fukaya categories of symplectic manifolds and dg enhancements of the bounded derived category of varieties.

The most concrete definition of Hochschild homology $HH_\ast(\mathcal{A},\mathcal{A})$ is via the cyclic bar complex. One takes the direct sum over all $d\geq 0$ and all sequences of objects $X_0,\dots, X_d$ of the tensor product
$$ \hom(X_d,X_0) \otimes \hom (X_{d-1},X_d) \otimes \dots \otimes \hom(X_1,X_2) \otimes \hom(X_0,X_1). $$
You should picture this tensor product not as a linear chain but as circular one; the term $\hom(X_d,X_0)$ is special. In the case where $\mathcal{A}$ is the Fukaya category $\mathcal{F}(M)$ of a symplectic manifold $M$, the $X_i$ are (decorated) Lagrangian submanifolds, and when these are transverse the elements of $\hom(X_i,X_j)$ are linear combinations of intersection points between $X_i$ and $X_j$. So the Hochschild chain complex has a basis given by cyclic sequences of intersection points, one of them marked as special.

The boundary operator is given by taking some sequence of $k\geq 1$ cyclically adjacent terms in the cyclic tensor product and composing them via one of the $A_\infty$-structure maps $\mu^k$ so as to shorten the cyclic sequence by $k-1$. In the Fukaya categorical case, the $\mu^k$ count pseudo-holomorphic $(k+1)$-gons. One does this in all possible ways and sums with hard-to-fathom signs as in Abouzaid's paper
1001.4593
(it would be wonderful if someone can tell me how to make these signs transparent). There is also a chain-lengthening contribution to the complex from the obstruction cochain $\mu^0$.

This concrete description has some real advantages; for instance, as Seidel noticed, there is a geometric description of a homomorphism from Hochschild homology to quantum cohomology

$$ HH_{\ast}(\mathcal{F}(M),\mathcal{F}(M)) \to QH^{\ast}(M) $$

(this for closed $M$) which is expected to be an isomorphism.

For computations, two facts are noteworthy. First, Hochschild homology has Morita-invariance properties. For example, it is unchanged under passing to the category of twisted complexes, which is useful because one can restrict attention to some collection of objects that generate the derived category. Second, it is the derived tensor product of graded bimodules (see Sasha's answer), which means in practical terms that you can compute it using much smaller complexes than the cyclic bar complex.

The conjecture that $HH_{\ast}(\mathcal{F}(M),\mathcal{F}(M))\cong QH^{\ast}(M)$ is consistent with mirror symmetry. In that case, the twisted complexes on $\mathcal{F}(M)$ (technically, the idempotent completion thereof - this doesn't affect $HH_\ast$ either) are quasi-equivalent to a dg-enhanced bounded derived category on the mirror manifold $W$, defined over some non-archimedean Novikov-type field.

Here my understanding is rather feeble, but I think the story is that $HH_\ast$ for this dg category is isomorphic to Hochschild homology *of the non-singular variety* $W$ (for various equivalent definitions, see Swan's article). This is known to be isomorphic to sheaf cohomology $H^\ast(W, \Omega^\ast_W)$ of the algebraic differential forms, hence to ordinary cohomology of $W$, hence finally to cohomology of $M$. So, if you have an HMS theorem for $M$ and $W$, you at least know that $HH_\ast(\mathcal{F}(M),\mathcal{F}(M))$ is isomorphic to $QH^\ast(M)$.