The rank (or order) of a tensor is a non-negative number 0, 1, 2, 3. . . that indicates its dimensionality.

A rank 0 tensor is just a plain number (scalar) like $a=1$. It has no dimensions and is like a geometric point which has position but no size.

A rank 1 tensor is a vector like

$$V_i=\begin{pmatrix}v_1\\v_2\end{pmatrix}$$

It has one dimension whose size in this instance is two and is like a geometric line of length two units. The rank one tensor have exactly one index which we label as $i$.

A rank 2 tensor is a matrix like

$$A_{ij}=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{pmatrix}$$

It has two dimensions whose sizes in this instance are three and two and is like a rectangle of length three and width two. A rank two tensor has two indices that we label $i$ and $j$.

A rank three tensor is shown in the diagram below.

It has three dimensions whose sizes here are four, three and two. It is like a cuboid of length four width three and height two. You can imagine that the cuboid contains six numbers which are components of the tensor. We label the tensor with three indices $i$, $j$ and $k$.

When we come to a tensor of rank 4 which we write with four indices as $T_{ijkl}$. We picture it as follows. Suppose each if the four dimensions is of size 4. The any one of the index may be a number from the set {1, 2, 3, 4}. Suppose we have $i=1$, $j=3$, $k=2$ and $l=3$ then this gives us the component $T_{1323}$ which is just a number. How many components does this tensor have? Since there are four choice for each of the four indices we have

$$4\times4\times4\times4=256\text{ components}$$

In general if we have a tensor of rank $r$ in a space of dimension $d$ then each of the $r$ indices can take $d$ values and we have

$$d\times d\times d\times\cdots\times d\text{ r times}=d^r\text{components}$$

Hence the number of components is dimension to the power of rank. Hence a tensor of rank 3 in the four-dimensional Minkowski space of relativity has 64 components.