SSRN displays various metrics about papers, authors, and institutions. On the paper level, you can see how many times its abstract has been viewed and how many times it has been downloaded. On the author level, you can see how many papers that author has posted, how many times each has been downloaded, and how that author ranks. And on the institution level, you can see the authors affiliated with an institution and their various download stats, as well as rankings.
To see some of the metrics, you will need to log in, but remember that registering is free.
For example, when you look at the abstract for Designing Islamic Constitutions: Past Trends and Options for a Democratic Future by Clark Lombardi, you see that the abstract has been viewed 12,018 times, the paper has been downloaded 6,938 times, and that download counts ranks it 662nd among papers on SSRN. (This snapshot was Jan. 14, 2018.)
For some papers, you will also see figures for references and citations. For example, Reconciling Efficient Markets with Behavioral Finance: The Adaptive Markets Hypothesis, by Andrew W. Lo 116 references and 29 citations. That means that the paper has 116 references in its bibliography and that 30 other papers on SSRN have cited it. (Snapshot on Jan. 14, 2018.)
These counts are generally not helpful for papers in law. Why? Because the automated system SSRN uses to generate them looks only at references that are listed at the end of a paper. Law journals (and legal scholars) almost always put citations in footnotes and do not have separate lists of references at the ends of papers. If you rely on these metrics, it will look as though a legal scholar has cited nothing and is never cited.
For tools to find and count citations to articles, see Tracking Citations to Articles.
You can slice the numbers in different ways: new downloads, number of new papers, new downloads per paper, total downloads, etc.
SSRN now has over 360,000 authors. To offer a little ego boost, SSRN periodically sends messages to the top 30,000 authors, by new downloads and by all-time downloads, to let them know they're in the top 10%.
SSRN ranks law schools by different measures—e.g., downloads in the last 12 months, all-time downloads, number of authors, downloads per author.
You can view the Top 750 Law Schools, the Top 350 U.S. Law Schools, or the Top 500 International Law Schools. ("International" here means "non-U.S.") You can change the sort by clicking on the heading at the top of a column.
You can search for a school, e.g., University of Washington. On Jan. 1, 2018, UW Law was ranked 25th in new downloads.
You can click on Authors to see a list of all the authors included with that institution. For UW Law (as for other schools) authors can include students as well as faculty. You can sort alphabetically, by recent downloads, by all-time downloads, by number of papers, and so on.
Bernard S. Black & Paul Caron, Ranking Law Schools: Using SSRN to Measure Scholarly Performance, 81 Indiana L.J. 83 (2006)
Gary Lucas, Jr., Measuring Scholarly Impact: A Guide for Law School Administrators and Legal Scholars, 165 U. Pa. L. Rev. Online 165 (2017)
Note that the metrics SSRN offers are limited.