Stanford Network Analysis Project (SNAP) has a website http://snap.stanford.edu/data/links.html with links to many network datasets (large and small).

Mark Newman is one of the most recognized network scientists who maintains a number of network datasets at his site http://www-personal.umich.edu/~mejn/netdata/.

Albert-László Barabási is one of the foremost network scientists and runs the Barabasi Lab at Northeastern University. More information about him can be found at http://barabasilab.com/rs-netdb.php.

The Gephi wiki has many network datasets as well as a number of files that are already in various proprietary formats. These can be found at https://wiki.gephi.org/index.php/Datasets.

KONECT is the Koblenz Network Collection available at the University of Koblenz-Landau. The University's site houses one of the best collections of network data, and provides easily accessible symbols that identify specific network attributes, which can be found at http://konect.uni-koblenz.de/.