The BFSS matrix model (Banks-Fischler-Shenker-Susskind 96) is the description of the worldline dynamics of interacting D0-branes. In the limit of large number $N$ of D0 branes supposed to encode the strong coupling limit of type IIA string theory (see also at M-theory).
The BFSS matrix model was argued to arise in several equivalent ways:
as the worldvolume theory of a large number of D0-branes in type IIA string theory,
as the Kaluza-Klein compactification of 10d super Yang-Mills theory to zero space dimensions,
as a certain non-commutative regularization of the worldvolume theory of the M2-brane in M-theory (Nicolai-Helling 98, Dasgupta-Nicolai-Plefka 02).
In any case, it ends up being a quantum mechanical system whose degrees of freedom are a set of 9+1 large matrices. These play the role of would-be coordinate functions and their eigenvalues may be in interpreted as points in a spacetime thus defined.
In the 90s there was much excitement about the BFSS model, as people hoped it might provide a definition of M-theory. It is from these times that Edward Witten changed the original suggestion that “M” is for “magic, mystery and membrane” to the suggestion that it is for “magic, mystery and matrix”. (See Witten’s 2014 Kyoto prize speach, last paragraph).
There is also the IKKT matrix model, which takes this one step further by reducing one dimension further down (D(-1)-branes). See also at membrane matrix model.
The original article is
The interpretation of the BFSS model as a regularized M2-brane worldvolume theory is discussed in
Hermann Nicolai, Robert Helling, Supermembranes and M(atrix) Theory (arXiv:hep-th/9809103)
Arundhati Dasgupta, Hermann Nicolai, Jan Plefka, An Introduction to the Quantum Supermembrane, Grav.Cosmol.8:1,2002; Rev.Mex.Fis.49S1:1-10, 2003 (arXiv:hep-th/0201182)
Further perspective includes
A review of further developments is in
Relation to the 6d (2,0)-supersymmetric QFT: