What are backlinks and why have them?

Back-links are links from other sites linking to a page or article on your site. Having back-links from other sites within your interest or industry is important to search engines because it tells them your site is so relevant to your subject that others use you as a resource.
For example if you're a plumber and sell or provide a specific plumbing product. Having the manufacture offer a back-link as a supplier of that product will help increase your search engine ranking. If you belong to a plumbing Union or Organization having a link from their site your yours would also help with rank authority.
Note: Be careful with back-links they can get you blacklisted from search engines like Google if you start a back-link chain campaign. In time if you own a website you'll at some point receive an email (spam) that will be read something like "Get 100 pr7 back-links to your site" where links are created on sites that are irrelevant to your subject or business, search engines weigh back-links heavily so when they see lots of links to your site that have nothing to do with your business they will remove your site from their search results. It is hard to get a site off of the blacklisted site list when this happens.