In MATLAB (and Julia?), a “mass matrix” is a matrix M in an implicit differential equation
M(x;t)\frac{\mathrm{d}x}{\mathrm{d}t} = f(x,u;t)
When M is rank deficient, this is effectively a DAE. Originally, this was the way to specify DAEs in MATLAB.
Sometimes, such models can be written in semi-explicit form as
\frac{\mathrm{d}x_1}{\mathrm{d}t} = f_1(x_1,x_2,u;t) \\
0 = f_2(x_1,x_2,u;t)
which may be simpler to write down. But perhaps the version with mass matrix is more efficient to solve (??).