I can write down matrices like:
$$ A = \left[\begin{array}{c c} 1 & 0 \\ 0 & 1 \end{array}\right]$$
How would I write down tensors nicely on paper?
For example, how would I notate a (1,2) tensor for 2-d vectors?
I heard one representation is as matrices.
Partitioned matrices natural here (i.e. matrices whose entries are themselves matrices rather than merely being scalars!). Also, see the Kronecker product for how the tensor product appears in this form.
For example, bilinear forms really shouldn't be written the way they usually are; the coordinate form of the dot product ought to be
$$ [ \begin{matrix} 1 & 0 &|& 0 & 1 \end{matrix} ] $$
The usual way to express a bilinear form as a matrix is not that of the bilinear form itself, but the result of transposing one of its horizontal dimensions. i.e i.e. if $B_{ij}$ are the components of the bilinear form, the matrix one usually writes down is the components of $\delta^{ik} B_{kj}$. (or $g^{ik} B_{kj}$ if you are using some other metric $g$)
Note that you can compute the product with a (column) vector along either of the horizontal dimensions: i.e. if $u,v$ are row vectors and $w$ a column vector, then we could take either
$$ [u | v] w = [ u w | v w ]$$ $$ [u | v]\left[ \begin{matrix}a \\\hline b \end{matrix} \right] = u a + v b $$
This extends to any sort of nesting of partitions. The main thing to keep in mind is that vectors should have a column of components (i.e. its index selects a row), and covectors should have a row of components (i.e. its index selects a column).