Nodes can have different weights (or are more important, or strategic). We can connect them using varied algorithms (for example connecting them to emphasize their weight (or importance, or strategic value). Then we can model graph.
It can be used to generate output (for example for search phrase [for example for search engine] or output language).
Hint: Tree can be modelled from graph by pulling one node and making it root (It will retain connections between siblings, i'd call it Basting Tree). You can imagine it.
Usual (more proper) definition of weighted graph is to associate a label (weight) with every edge in the graph. When using these words in this blog, i will always precise if graph associates weight(s) with nodes, edges, or both.