Citations are mentions of your business on the internet, with or without a link. They can be any combination of your company name, phone, address, zip code. Structured citations are the most common and are usually found on business listing sites. Unstructured citations are mentions of your business on blogs, event listings, newspapers sites, job databases, etc.
Citations are a major component to improving your local search visibility and not all citations are created equally. A mention of your business on a random spammy site can actually hurt your ranking, while having your business name, address and phone listed on whitehouse.gov gives you an incredibly high level of credibility.
It is important to check the quality of the citation source, before seeking it. A good resource for determining if you want to seek a citation is Whitespark, which has an excellent blog post on identifying quality citation sources.