A vector is an array of scalar numbers. We can identify each individual number by its index in that ordering. Typically, we give vectors lowercase names in bold typeface, such as $\mathbf{x}$.
A matrix is a $2$-dimensional (2D) array of numbers. This means that every element in the matrix is identified by two indices, commonly $i$ representing the row-index and $j$ representing the column-index.
In the context of Machine Learning, it is convenient to think of a tensor as an $n$-dimensional array. Tensor dimensionality is also commonly referred to as its order, degree or rank, which formally is the sum of the tensor contravariant and covariant indices.
