Any website designed to allow multiple users to publish content themselves. The information may be on any subject and may be for consumption by (potential) friends, mates, employers, employees, etc. The sites typically allow users to create a "profile" describing themselves and to exchange public or private messages and list other users or groups they are connected to in some way. There may be editorial content or the site may be entirely user-driven. Content may include text, images (e.g. http://flickr.com/), video (e.g. http://youtube.com/) or any other media.
Social networks on the the web are a natural extension of mailing lists and buletin boards. They are related to wikis like http://wikipedia.org/ but typically do not allow users to modify content once it has been submitted, though usually you can publish comments on others' submissions.
Different sites have different emphasis. For example, http://friendsreunited.co.uk/ (one of the earliest such sites) focusses on listing former acquaintances; http://myspace.com/ is music-oriented; http://linkedin.com/ aims to connect business partners; http://del.icio.us/, http://stumbleupon.com/ and http://digg.com/ are for exchanging links to favouirite websites. There are many more.
Sometimes the social aspects are a side-effect of bringing together people with shared interests, e.g. http://slashdot.org/ (IT), other times they become more important than the original purpose, e.g. http://worldofwarcraft.com/ (fantasy gaming).