Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb{C}$, and $\mathfrak h$ a Cartan subalgebra. Let $f: \mathfrak g \rightarrow \textrm{End}(V)$ be a representation of $\mathfrak g$. For $\lambda \in \mathfrak h^{\ast}$, let
$$V_{\lambda} = \{v \in V : f(H)v = \lambda(H)v \textrm{ for all } H \in \mathfrak h \}$$
be the weight space of $\lambda$. The nonzero elements of a weight space are called weight vectors.
The following picture is from the Wikipedia article https://en.wikipedia.org/wiki/Weight_(representation_theory). It is captioned, "Example of the weights of a representation of the Lie algebra $\textrm{sl($3,\mathbb{C}$)}$.
I don't understand what the picture is getting at. I think this representation is the differential of the adjoint representation $\textrm{Ad}: \textrm{SL}_{3,\mathbb{C}} \rightarrow \textrm{GL}( \textrm{Lie}(\textrm{SL}_{3,\mathbb{C}}))$. I feel like I should know this (or maybe I already know this). I have a decent understanding of reductive groups over algebraically closed fields, so perhaps I can relate the picture to what I know about that somehow.