They are ways of storing numbers and other things. Typical matrix has rows and columns. Actually called a 2-way matrix because it has two dimensions. Of course, you might collect the same data on the same people at 5 points in time. Notice that the rows and columns point to different class of entities.
Rows are people, and columns are properties.
Each different class of entities in a matrix is known as a mode. Suppose we had a different matrix. Say we recorded the number of times that each of you spoke to each other of you. A 1-way matrix is also called a vector. If it has rows but just one column, its a column vector.