According to its creator, "Publish or Perish is a software program that retrieves and analyzes academic citations. It uses Google Scholar and (since release 4.1) Microsoft Academic Search* to obtain the raw citations, then analyzes them." You can read much more about it, including a comparison of its results with ISI Web of Science results, on the Publish or Perish web site.
Publish or Perish is freely available online. You must download and install it on your computer to use it.
*As of Summer 2014, there appears to be some question as to whether Microsoft Academic Search is still operating. Updates as they become available.
Many free citation search engines are discipline-specific. The majority of them are in the sciences. A partial list is below. (To suggest additions, please contact Eloise Flood using the profile box on this page.) The links will take you to the home pages of each service.
ArXiv: Hosted by Cornell University. ArXiv is a venerable open-access repository and search engine for preprints of scholarly papers in the sciences. Its primary focus is physics, but it also includes other sciences, mathematics, and statistics papers. ArXiv uses INSPIRE (see below) as its citation searcher.
ChemXSeer: Hosted by Penn State University. ChemXSeer is an open-access repository and search engine for papers in chemistry.
CiteSeerX: Hosted by Penn State University. CiteSeerX is an open-access repository and search engine for papers in computer and information science. The interface is easy to use and citation searching is a breeze.
INSPIRE: The successor to SPIRES, a high-energy physics repository and search engine created by Stanford University. INSPIRE launched in 2012. INSPIRE has very good citation searching tools.
RePEc: A decentralized database that offers open access to bibliographic data and/or full text for more than 1.2 million scholarly papers on economics. Excellent citation analysis can be found by clicking the CitEc link.
SSRN: Produced by SSEP, an independent company, and overseen by an impressive board of trustees and directors drawn from academic institutions. Social Sciences Research Network provides access to bibliographic data and/or full text for scholarly papers across the social sciences. To view citation data, you must register (free).