This \mathbf{I}=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix} is what we call an identity matrix: anything we multiply it my stays the same. This particular one is the \left(2\times2\right) identity matrix. But since \mathbf{IM}=\mathbf{MI}=\mathbf{M} (order doesn't matter for identity multiplication, we can figure out what \mathbf{I} must have the same number of rows and columns, since \mathbf{IM} has the same number of rows as \mathbf{I}, and \mathbf{MI} the same number of columns.

Thus \mathbf{I} is what we call a square matrix, meaning it has order \left(n\times n\right). As for what's inside the identity matrix, remember how matrix multiplication works:

We dot product each row i of the first matrix with each column j of the second matrix to produce the entry ij.
If we want \mathbf{IM}=\mathbf{M}, that means that the i^{\th} row of \mathbf{I} dotted with the j^{\th} column of \mathbf{M} must give\mathbf{M}_{ij}.
The j^{\th} column of \mathbf{M} contains \mathbf{M}_{ij} in the i^{\th} row, and we want to take only that value when we do the dot product.
We can achieve that by having a 1 in the i^{\th} column of the i^{\th} row of \mathbf{I}, and 0's in the other columns.

The i^{\th} row of the i^{\th} column is the diagonal from the top left to the bottom right:

\begin{pmatrix}1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 & 0\\ \cdots & \cdots & \ddots & \cdots & \cdots\\ \cdots & \cdots & \cdots & 1 & \cdots\\ 0 & \cdots & \cdots & 0 & 1\end{pmatrix}

Identity, Inverses & Determinants