Given a 2-Segal space $X_\bullet$ such that the spans
and
admit pull-push integral transforms in some given cohomology theory $h$. Then the Hall algebra of $X$ with coefficients in $H$ is the associative algebra structure on $h(X_1)$ induced by these pull-push operations.
This is the perspective of Dyckerhoff-Kapranov 12, def. 8.1.8.
Specifically for a given algebraic stack $X$ and with
denoting the moduli stack of 2-flags of coherent sheaves on $X$, then the corresponding pull-push multiplication on the motivic Grothendieck ring $K(X)$ is called the motivic Hall algebra of $X$ (due to Dominic Joyce reviewed e.g. in Bridgeland 10, 4.2). Discussion of motivic Hall algebras of Calabi-Yau 3-folds is in (Kontsevich-Soibelman 08).
The Hall algebra of an abelian category is the Grothendieck group of constructible sheaves/perverse sheaves on the moduli stack of objects in the category. The Hall algebra is an algebra because the constructible derived category of the moduli stack of objects in an abelian category is monoidal in a canonical way.
This perspective is taken from (Webster11). See there for more details.
A good survey is given in
The characterization via 2-Segal spaces is due to
Canonical references on Hall algebras include the following.
M. Kapranov, Eisenstein series and quantum affine algebras, Journal Math. Sciences 84 (1997), 1311–1360.
M. Kapranov, E. Vasserot, Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000), 565-576, arxiv/9812016
Bernhard Keller, Dong Yang, Guodong Zhou, The Hall algebra of a spherical object, J. London Math Soc. (2) 80 (2009) 771–784, doi, pdf
C. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591.
O. Schiffmann, Lectures on Hall algebras, arXiv:math/0611617
O. Schiffmann, E. Vasserot, The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, arXiv:0802.4001, (2008), to appear in Compositio Math.; The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A2, arXiv:0905.2555
O. Schiffman, Drinfeld realization of the elliptic Hall algebra, arxiv/1004.2575
O. Schiffmann, E. Vasserot, Hall algebras of curves, commuting varieties and Langlands duality, arxiv/1009.0678
B. Toen, Derived Hall algebras, arxiv/0501343
Alexander Efimov, Cohomological Hall algebra of a symmetric quiver, arxiv/1103.2736
Description of seminar on stability conditions, Hall algebras and Stokes factors in Bonn 2009 (D. Huybrechts), pdf
wikipedia: Hall algebra, Ringel-Hall algebra
sbseminar blog: Hall algebras and Donaldson-Thomas invariants-i
Bangming Deng, Jie Du, Brian Parshall, Jianpan Wang, Finite dimensional algebras and quantum groups, Mathematical Surveys and Monographs 150, Amer. Math. Soc. 2008. xxvi+759 pp. (chap. 10: Ringel-Hall algebras) MR2009i:17023)
David Hernandez, Bernard Leclerc, Quantum Grothendieck rings and derived Hall algebras, arxiv/1109.0862
Parker E. Lowrey, The moduli stack and motivic Hall algebra for the bounded derived category, arxiv/1110.5117
Motivic Hall algebras:
Tom Bridgeland, An introduction to motivic Hall algebra (arXiv:1002.4372)
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (arXiv:0811.2435)
M. Kontsevich, Y. Soibelman, Motivic Donaldson-Thomas invariants: summary of results, arxiv/0910.4315
Maxim Kontsevich, Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, arxiv/1006.2706