The most important use of matrices lies in representing linear transformations on a vector space.
A matrix represents the tranformation which takes the first basis vector into first column of the matrix, second basis vector into the second column of the matrix, j-th basis vector into j-th column.
What does it do to other vectors?
Remember that any other vector, say v can be expressed as a linear combination of the basis vectors: v gets transformed by the transformation to that same linear combination of the column vectors of the matrix.
For example, the sum of the first two basis vectors gets mapped into the sum of the first two columns of the matrix; the average of the two basis vectors into the average of the columns, and so on.
Notice that the same transformation acting on vectors will usually be described by a different matrix if you use a different basis.