This question aims at providing links to definitions of graphs that either come from real-world problem or research that can be accessed and used freely; good examples in that vein are TSPLIB95 or the DIMACS Implementation challenges.
Besides the link answers should contain a description of the graphs that are defined like e.g. their origin, graph-theoretic properties and possible uses e.g. graph-theoretic problems for which the data can be used as testcases or benchmarks.
A very extensive collection of graphs from real world problems is the Stanford Large Network Dataset Collection. The screenshot shows the graphs in the category "social networks", there are a dozen of other categories.
The house of graphs https://hog.grinvin.org/ contains a large number of graphs that are interesting for researchers in graph theory, with relevant graph parameters computed for each of the graphs (so you can also search for graphs satisfying a number of specific properties).
The meta-directory https://hog.grinvin.org/MetaDirectory also contains links to large lists of graphs of specific classes (for instance all planar graphs on at most 11 vertices...)
In the same vein, I would also recommand the Konect database: http://konect.cc/networks/ or networkrepository: https://networkrepository.com/
