Backlinks are incoming links to a website or web page. The number of backlinks is an indication of how popular or important that website or webpage actually is.
Search engines, such as Google often use the number of backlinks that a website has as one of the factors for determining that website's search engine ranking. For example, Google's Page Rank algorithm uses backlinks to help determine a site's rank, the higher your sites Page Rank the better. To view your Google page rank you can either use Google toolbar or visit a site such as this.
Websites often employ various techniques (called search engine optimization) to increase the number of backlinks pointing to their website. When linking to other sites it is better to link to sites with a good page rank as this will have a positive impact on your own page rank, where as linking to lots of low page ranking or zero page rank sites will have a negative impact on your own page rank (see link farms for more information)