FTP (file transfer protocol) is a method to transfer files between computers, including between your computer and a web server.  An FTP program (client) does this for you, looking something like a file manager that you'd use on your own computer to move files around.
Free FTP programs are available for:


Not free, but my favorite full-featured file manager. Includes FTP and Zip.
Total Commander
or a dedicated FTP program:


(Most Linux distros have an FTP program included)

Programs for $


Not free: 
Shown below (in Filezilla and Total Commander) are how a typical (more complicated) site is laid out.  A Filezilla tutorial is posted here.
Filezilla Screen
To transfer files, just drag them from one screen to another.  If the file is in a sub-folder, you'll have to open the sub-folder on the web server.
To copy and/or download files from the web server to your local computer, drag them in the opposite direction.
Total Commander Screen
Total commander

Troubleshooting Links and Pictures