Under a rotation, coordinates transform like
where $R$ is a rotation matrix. (A repeated index like $j$ here is implicitly summed over in this index notation.)
A vector (also known as a tensor of rank 1) consists of 3 components that transform in the same way,
A tensor of rank 2 consists of 9 components that transform like the product $v_iv_j$ of two vectors:
The moment-of-inertia tensor has this transformation law, which explains why it is called a tensor of rank 2 rather than simply a matrix. A matrix is just a square array of numbers with no particular transformation law under coordinate transformations.
A tensor of rank 3 consists of 27 components that transform like the product $v_iv_jv_k$ of three vectors:
and so on for any rank. So there are higher-rank tensors which don’t look like matrices.
There are fancier (and better) ways to think about tensors as more than just a bunch of components, but thinking about how their components transform under a coordinate transformation is a common first intro to them.
The tensor concept then generalizes to higher-dimensional spaces such as 4D spacetime; to other transformations besides rotations, such as Lorentz transformations; and to curved manifolds rather than flat Euclidean space and Minkowski spacetime. If you continue studying physics you are likely to encounter all of these kinds of tensors.