FTP stands for File Transfer Protocol and, as the name implies, it is a way of transferring files between computers. It is also one of those wonderful computer terms that's both a noun and a verb. As a noun, FTP is the name for a method of sending files, but also the name of the program that actually sends the files.
Uploading involves transferring files from a personal computer to a server. Downloading involves transferring a file from a server to a personal computer. FTP uses TCP/IP (Transmission Control Protocol/Internet Protocol) to transfer your files.
To enhance security, many companies now use sFTP or FTP/S, the "secure" forms of FTP, believing that data traveling across this protocol is safe. It's true that secure forms of FTP have additional encryption while commands and data are in transit across the Web.
Dropbox might be faster when you send larger quantity of files. FTP is as fast as you can get when we talk speed but it takes too much "talk" between server and client computer for each file, so the ftp seems to be slower. As there is no real file-content transfer, this has to be faster than anything else.