Some of our more technically-oriented people have helped me as I try to upload new data to the site. They've told me that I need to download something called an FTP client.

FTP stands for "File Transfer Protocol". But why is it called a client?

It acts as a client to the FTP server.

A server is a program that offers a service. A client is a program that requests a service. These days you have many clients running on your computer: e.g., a web client (aka browser), mail client, etc. These server-client pairs all use protocols: e.g., FTP, SMTP (standard mail transfer protocol), HTTP (hypertext transfer protocol), etc. All of these protocols are what provide most of the functionality of the Internet. They are defined in de facto standards functional specifications called RFC (request for comments) papers. Sometimes the term server is used not for the software, but for the hardware. I can refer to some specific computer as the mail server, when what I mean is the host running the SMTP server software (traditionally called a daemon in Unix).