Paper was added to Alpha in the Seecret Friday 6 (v1.0.11) patch, and is used the same way in Beta. Paper can be crafted from Sugar Cane. Paper currently serves no gameplay purpose, except to make books which are used to craft bookshelves for decoration. Adding functionality in the form of writing them, similarly to signs, remains a popular request, however. Several mods exist to serve this.
It is likely that it will be used in the crafting of maps, which are currently in development.
Paper is currently made from sugar cane. In ancient times it was made from the papyrus plant. Currently it is produced from wood pulp. It is unclear whether sugar cane will remain the required material to create paper or if Notch will introduce or change this requirement.