It would be impossible to list all datasets relevant to political science research. However, here are some of the more commonly used ones to get you started in your research:
Unified Democracy Scores – the latest in measuring democracy. Very fancy.
State of the USA This is not complete yet, but when it is, this will be really cool.
The Macro Data Guide Not a bad place to start if you are unfamiliar with available datasets
Online Africa – ICT data for Africa
Paul Hensel’s International Relations Data Site (lots of links)
Brancati’s Constituency-Level Elections dataset
Hyde and Marinov’s National Elections Across Democracy and Autocracy dataset
Kollman et al’s Constituency-Level Elections Archive
Amnesty International torture data
European Protest and Coercion dataset
Election quality and monitoring data
Bear Braumoeller’s data webpage This one is more than just data; also info on data tools like visualization, etc.
American Legislatures dataset