Let $(V,\Phi)$ be a root system. I have a question about the definition given below (Borel, Linear Algebraic Groups):
There is no further explanation given as to what these "weights" are. What else is there to a Dynkin diagram besides the vertices $\Delta$ and the number $n_{\alpha,\beta}n_{\beta,\alpha}$ of edges joining any two vertices $\alpha, \beta$? Here $n_{\alpha,\beta}$ is the unique integer such that $$r_{\alpha}(\beta) = \beta - n_{\beta,\alpha}\alpha$$ I know there is more to the definition because otherwise the diagram is a Coxeter diagram, which does not determine the root system up to isomorphism.
Also, why is $n_{\alpha,\beta}n_{\beta,\alpha}$ always a nonnegative integer?
Also, what is meant by a morphism of Dynkin diagrams? What do we mean by functorial in $(\Phi,\Delta)$?
Let $\Phi$ be an irreducible root system of rank $r$. The Dynkin diagram will have $r$ vertices, and although Dynkin diagrams of type $A_r$ or $D_r$ can be seen deduced from the Coxeter graph, types $B_r$ ($r\geq 2)$ and $C_r$ ($r\geq 3)$ cannot be deduced from the Coxeter graph.
Consider $B_3$ and $C_3$. These both have Coxeter diagram:
$$\circ -\circ\stackrel{4}-\circ,$$ despite having different Dynkin diagrams and Cartan matrices, $$\begin{pmatrix}2&-1&0\\-1&2&-2\\0&-1&2 \end{pmatrix},\quad \begin{pmatrix}2&-1&0\\-1&2&-1\\0&-2&2\end{pmatrix},$$ for $B_3$ and $C_3$ respectively.
Where we can distinguish between these two via the Dynkin diagrams respectively by adding an arrow pointing pointing towards the smaller of a pair of repeated roots:
The entries of the Cartan matrices are called Cartan integers, given by $(M)_{ij}=\langle \alpha_i,\alpha_j\rangle$ where there is a fixed order on the simple roots $\{\alpha_1,\cdots,\alpha_r\}$. In this language, the number of edges joining the $i^{\text{th}}$ vertex to the $j^{\text{th}}$ vertex is given by $\langle \alpha_i,\alpha_j\rangle \langle \alpha_j,\alpha_i\rangle$, which must be nonnegative by the multiplication of two Cartan integers.
Now we can see in the Coxeter graph case, we are determining the entire Dynkin diagram, meaning all roots have the same length, since in such a case $\langle \alpha_i,\alpha_j\rangle = \langle \alpha_j,\alpha_i\rangle$, but as we can see from the Cartan matrices, this fails in $B_3$ and $C_3$ above, so the arrow pointing to the shorter of the roots is needed to completely describe the two, recovering all Cartan integers.
Note: when $\Phi$ is an irreducible root system, there are only two root lengths in $\Phi$. So we are safe to talk about short and long roots.
Attempt short answer - A Dynkin diagram can be built from the Cartan matrix(or from the root system itself), where each simple root is a vertex in the graph. When there is only one line connecting two vertices if the two roots have the same length. An arrow can be used to indicate that the two roots have different lengths, with the note above, one small and one long, pointing from the longer to the shorter root.