FTP is an acronym for File Transfer Protocol. It is a "protocol" (or "set of rules") which describes how to tell an FTP server to do things like give directory listings or upload and download files.
Note that you would usually have an FTP client (like RFtp PRO) actually doing the communicating with an FTP server (and worrying about those "rules")!
An FTP server manages (controls access to) a set of files and directories. Depending on the access rights of the currently connected user, it can offer directory listings, upload and download files, and perform maintenance services including creating, renaming, and deleting files and directories.
An FTP client communicates with (talks to) an FTP server on behalf of a user, assembling and transmitting the commands for displaying directories, transferring files, and maintenance operations.
In addition, it should present these transactions (and their results) in the context of a "friendly" and familiar user interface.
An FTP proxy server "sits between" an FTP client and an FTP server. Its job (usually for reasons of security and/or performance) is to pass requests from the client to the server and return the results back to the client.