Citations are defined as mentions of your business name and address on other webpages—even if there is no link to your website. An example of a citation might be an online yellow pages directory where your business is listed, but not linked to. Citations can also be found on local chamber of commerce pages, or on a local business association page that includes your business information, even if they are not linking at all to your website.
Citations are a key component of the ranking algorithms in Google and Bing. Other factors being equal, businesses with a greater number of citations will probably rank higher than businesses with fewer citations.
Citations from well-established and well-indexed portals (i.e., Superpages.com) help increase the degree of certainty the search engines have about your business's contact information and categorization. To paraphrase former Arizona Cardinals' coach Dennis Green, citations help search engines confirm that businesses "are who we thought they were!"
Citations are particularly important in less-competitive niches, like plumbing or electrical, where many service providers don't have websites themselves. Without much other information, the search engines rely heavily on whatever information they can find.
Citations also validate that a business is part of a community. It's hard for someone to fake membership in a chamber of commerce or a city or county business index, or being written about in a local online newspaper or popular blog.
Citations and links from these kinds of websites can dramatically improve your local search engine rankings.