symmetric monoidal (∞,1)-category of spectra
The syntagm chiral algebra has two meanings: one in physics and a related but more specific one after the work fo Beilinson-Drinfeld. In QFT the chiral operator has two classes of eigenmodes left-handed and right-handed; they give rise to left and right chiral fields. CFT in 2d, in particular has two chiral parts expressed via holomorphic and antiholomorphic functions in OPEs.
Beilinson and Drinfel'd were unhappy with non-illuminating definition of vertex operator algebra and invented a mathematical definition of version of chiral conformal field theory on an algebraic curve, under the name chiral algebra; their manuscript has being circulating from around 1995 as a long preprint and being more recently published by Amer. Math. Soc.
The standard textbook is
An “example-driven digest” with applications in string theory is
Dennis Gaitsgory, Noted on 2d conformal field theory and string theory, from p. 1017 on in
Pierre Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and Edward Witten, eds. , Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
Generalization to higher algebra is discussed in