A matrix is a representation of numbers in a grid that can represent a variety of different mathematical concepts.
It can represent a linear transformation for example, in which case the columns of the matrix represent where the identity vectors i hat and j hat would end up in a new space, geometrically speaking.
Matrices are similar to functions in this way, where they have an input that can be another matrix or Vector, and have an output that is transformed. It also means that the matrix notation is most similar to function notation, in the sense that matrix transformations are applied right to left, much like functions are.