Freenet is free software which lets you anonymously share files, browse and publish "freesites" (web sites accessible only through Freenet) and chat on forums, without fear of censorship. Freenet is decentralised to make it less vulnerable to attack, and if used in "darknet" mode, where users only connect to their friends, is very difficult to detect.
Communications by Freenet nodes are encrypted and are routed through other nodes to make it extremely difficult to determine who is requesting the information and what its content is.
Users contribute to the network by giving bandwidth and a portion of their hard drive (called the "data store") for storing files. Files are automatically kept or deleted depending on how popular they are, with the least popular being discarded to make way for newer or more popular content. Files are encrypted, so generally the user cannot easily discover what is in his datastore, and hopefully can't be held accountable for it. Chat forums, websites, and search functionality, are all built on top of this distributed data store.
An important recent development, which very few other networks have, is the "darknet": By only connecting to people they trust, users can greatly reduce their vulnerability, and yet still connect to a global network through their friends' friends' friends and so on. This enables people to use Freenet even in places where Freenet may be illegal, makes it very difficult for governments to block it, and does not rely on tunneling to the "free world".
See also the Free software brochure :-)