What does "local" mean?
Local property - a property that relates to a specific vertex and its near neighbors. For example, the number of neighbors or 2nd order neighbors a vertex $v$ has. "locally looks like a tree" more formally could be written as the sub-graph of vertices with distance $k$ or less of the vertex $v$ is a tree.

Global property - a property that relates to the graph, like its diameter, average degree, number of connected components or even its size.