By default, FTP servers require the user to login in order to upload or download files. Many web hosts, however, additionally offer the ability to use anonymous FTP. It basically gives you the ability to allow users to login to the FTP server without providing a username or a password.
What it really does is give users access to a username called anonymous, which does not require a password. They can then perform whatever tasks that are allowed for that user. For example, the anonymous use may be allowed to download public files but probably will not be allowed to upload. Downloading with FTP takes load off of the web server and usually offers more connections so more users can access the files at once.
Despite its name, anonymous FTP is not completely anonymous. As a website owner, you can still track IP addresses and host name, in case someone attempts to compromise your system. If you do not offer public downloads of any sort, it is usually a good idea to disable anonymous FTP.