> **Slogan:** Hochschild homology is a (derived) categorification of the trace. 

This means the identity at the end of John Pardon's answer is a categorification of the identity $\text{tr}(AB) = \text{tr}(BA)$.

To see this the idea is to think of bimodules as categorifications of linear maps. One way to think about this is in terms of the <a href="https://ncatlab.org/nlab/show/Eilenberg-Watts+theorem">Eilenberg-Watts theorem</a>, which identifies $(A, B)$-bimodules with cocontinuous functors $\text{Mod}(A) \to \text{Mod}(B)$: every such functor is tensoring with some $(A, B)$-bimodule. Composition is given by composition of cocontinuous functors, or tensoring bimodules. 

A particularly nice special case is where $A = k^n, B = k^m$ for $k$ a field (and we consider bimodules over $k$): then $(A, B)$-bimodules are $m \times n$ "matrices" of vector spaces over $k$, and composition / tensor product is "matrix multiplication." This really makes the analogy to linear maps particularly explicit. For more details see <a href="https://qchu.wordpress.com/2016/05/31/higher-linear-algebra/">this blog post</a>. 

Bimodules form part of a 2-category whose objects are rings (or more generally algebras over a base commutative ring), morphisms are bimodules, and 2-morphisms are morphisms of bimodules. This 2-category is symmetric monoidal with monoidal product given by the tensor product of rings, which satisfies a universal property analogous to the universal property of the tensor product of vector spaces: namely, functors $\text{Mod}(A) \times \text{Mod}(B) \to \text{Mod}(C)$ which are cocontinuous in each variable can be identified with functors $\text{Mod}(A \otimes B) \to \text{Mod}(C)$ (and hence with $(A \otimes B, C)$-bimodules). 

Now, in any symmetric monoidal (higher) category one can define <a href="https://ncatlab.org/nlab/show/dualizable+object">dualizable objects</a> and traces of endomorphisms of dualizable objects; see <a href="https://qchu.wordpress.com/2012/11/06/string-diagrams-duality-and-trace/">this blog post</a> for details and pictures, which among other things explains what taking traces has to do with circles. In this case 

 * every ring $A$ is dualizable with dual $A^{op}$, 
 * endomorphisms correspond to $(A, A)$-bimodules $M$, and 
 * the trace turns out to be the zeroth Hochschild homology $HH_0(A, M)$. 

(In particular you can verify that when $A = k^n$, so that an $(A, A)$-bimodule is an $n \times n$ matrix of vector spaces, the trace is the direct sum of the diagonal entries, just as expected by analogy.) 

The full Hochschild homology is obtained by <a href="https://en.wikipedia.org/wiki/Derived_functor">deriving</a> this whole story, so that now morphisms are given by <a href="https://en.wikipedia.org/wiki/Derived_category">derived categories</a> of bimodules and composition is given by the <a href="https://stacks.math.columbia.edu/tag/09LP">derived tensor product</a>. 

All of this can be thought of as an elaboration of some particularly interesting special cases of 1-dimensional <a href="https://en.wikipedia.org/wiki/Topological_quantum_field_theory">topological field theory</a>, where traces correspond to circles, and higher-dimensional topological field theory features generalizations of Hochschild homology involving more complicated manifolds.