American National Election Studies (NES)
The National Election Studies series is one of the premier sources for data on voting behavior and political attitudes in the post-WWII United States. Upon registering (which is free), users can download data and SAS/SPSS/Stata program files and documentation for any of the various NES studies. The SDA Archive at UC-Berkeley also allows users to access recent versions of the NES and extract particular variables from them for those who have specific questions in which they are interested.
Congressional Quarterly Voting and Elections Collection
The CQ Voting and Elections Collection brings together a great deal of data on election results in the United States at both the federal and state levels. This resource is also available via Databases at Emory.
OpenSecrets is a watchdog group with an extensive clearinghouse of data on campaign contributions and lobbying. In addition to its main site, OpenSecrets also has an Open Data Initiative in which users can register to download bulk amounts of OpenSecrets' data for additional analysis.
Policy Agendas Project
UT-Austin's Policy Agendas Project is an ambitious undertaking that strives to "[collect and organize] data from various archived sources to trace changes in the national policy agenda and public policy outcomes since the Second World War." The data collections cover items such as Presidential State of the Union Addresses, roll-call votes and committee hearings in Congress, federal spending, and the priorities and "moods" of the public.
Roper Center for Public Opinion Research - Exit Polls
The Roper Center is one of the country's premier centers for polling data. Amongst (many) other items in its holdings, Roper has an extensive collection of exit polls for federal elections and some state elections. This resource is also available via Databases at Emory.
VoteView, which is a project of Dr. Keith Poole, provides freely-available Congressional data for roll-call votes, party polarization, and ideology scores. See https://legacy.voteview.com/ for data from earlier iterations of the project.