The National Archives and Records Administration manages libraries for each of the presidents from Hoover onwards. Records and papers relating to each president and their administrations are passed into the care of NARA at the end of the president's term of office, and are then held and made available to researchers at the relevant Presidential Library. For presidents from Reagan onwards, these records are subject to the Presidential Records Act and you are likely to need to request them under the Freedom of Information Act. This can be slow and will incur a fee; see the full NARA guide to FOIA requests for further information.
For a list of the NARA operated Presidential Libraries, please see below or the National Archives website.
For presidents prior to Hoover, the location and availability of papers varies considerably. The papers of most presidents are held at the Library of Congress, but other collections may be found in various libraries and archives throughout the United States. There are some dedicated "Presidential Libraries" for earlier presidents, often linked to a museum and historic site, however it is important to be aware that these are not part of NARA and the extent of their archival holdings varies.
Another key distinction to be aware of is between public and private papers. The public papers of the presidents are compiled and published in the following series, available in the Vere Harmsworth Library and also online in various sources (see the US History Sources guide for full details):
For many presidents, the private papers have been or are in the process of being collated and published, and if so, these should be available in the VHL; search SOLO. Others are available in microfilm collections or online - see the US History Sources guide for full details.
A good resource for locating collections of presidential materials is Connecting Presidential Collections, run by the University of Virginia's Miller Center.
The largest collection of presidential papers for presidents prior to Hoover may be found in the Manuscript Division at the Library of Congress. Collections of presidential papers may also be found in various other libraries and archives, and Presidential Libraries exist for several presidents (often connected with a museum and historic site), but these are not managed by NARA and the extent of their archival holdings varies.