In simple terms, a Local Citation is simply where your company is mentioned on other websites and places found on the Internet. Local citations are used heavily in helping you to rank in local search results.
An example of a citation could be a business directory such as Yell, Thompson Local or Brown Book where your company is mentioned explicitly by name. Local citations do not to include a link to your site. It could also be where your company is mentioned, cited, referenced or spoken about on other local websites.
What do they do?
Citations are used to help rank your Google+ Local map listing (this was formerly called Google Places Pages) by providing Google with credible sources of information about your business in order that Google can understand your business exists, is legitimate and that what you say about your business is true and accurate by allowing Google to corroborate what is said about your business from multiple sources. Here we see local search results, characterised by the balloon icons and the address or location of the business appearing next to the listing.