The origin of the Sugawara construction in Physics is, not surprisingly, the 1968 paper A field theory of currents by Hirotaka Sugawara. (There was also work of Sommerfield at about the same time.) The context of that work was to find a theory of strong interactions. (Recall that the currently favoured theory of strong interactions (QCD) would not be discovered until the mid 1970s.) There was a feeling at the time that there might not be a field theory describing the strong interactions, whose dynamical fields are associated to the particles themselves. (QCD is such a theory, of course.) Hence Sugawara's idea was to quantise the theory using "currents" (field bilinears) as the elementary dynamical variables. His theory is a four-dimensional theory and I think it is fair to say that it is now just of historical interest.
What most people call the Sugawara construction is what Marty Halpern insisted in calling the affine Sugawara construction. This is a construction in two-dimensional conformal field theory by which a Virasoro element is constructed in the conformal field theory associated to an affine Kac-Moody algebra. The construction has a long history of papers converging to the correct formula. I forgot the actual sequence of papers, but it probably starts with a 1971 paper of Bardakçi and Halpern, who introduced the construction, and ends with Knizhnik's and Zamolodchikov's 1984 paper. (But I stand to be corrected on this.)
A possibly physical interpretation, in the context of string theory, is that this is a quantisation of string propagation on a Lie group (with a bi-invariant metric), whose classical action is given by the Wess-Zumino-Witten model. In that model the dynamical fields are maps $g:\Sigma \to G$, where $\Sigma$ is a Riemann surface and $G$ is a Lie group with a bi-invariant metric. Just as in the original four-dimensional Sugawara construction, it is simpler to quantise the currents $g^{-1}\partial g$ and $\bar\partial g g^{-1}$ than the actual fields $g$. This was done by Edward Witten in his celebrated paper Non-abelian bosonization in two dimensions. The Sugawara construction serves to prove the exact quantum conformal invariance of the Wess-Zumino-Witten model to all orders in perturbation theory, something which is not possible to do by quantising the original fields $g$.
Mathematically, it embeds the Virasoro algebra in the vertex algebra of an affine Kac-Moody algebra (actually this fails at the so-called critical level) in such a way that any module of the Kac-Moody algebra (of noncritical level) is also a Virasoro module. If $\mathfrak{g}$ is a simple Lie algebra and $\widehat{\mathfrak{g}}_\ell$ the corresponding (untwisted) affine Kac-Moody algebra at level $\ell$, then the central charge of the Sugawara Virasoro element is given by $$c = \frac{\ell \dim\mathfrak{g}}{\ell + h^\vee}$$ where $h^\vee$ the dual Coxeter number. (So $\ell = - h^\vee$ is the critical level.)
The defining property of the Sugawara construction is that the currents are primary fields (of weight 1) of the Virasoro element.
This same construction extends to the affinisation of any metric Lie algebra; that is, admitting an ad-invariant metric.
