What does downloading off the Internet mean?

To download something off the internet means to transfer a file from a web server to your computer. For example, when you visited WikiAnswers, you downloaded an HTML file that told your browser how to display the page, as well as the images on the page. When you right click an image and select "Save As..." the subsequent act of saving it on your hard drive is considered a download.

When you're connected to the internet, your computer is almost constantly uploading (transferring files from your computer to a web server) and downloading (transferring files to your computer from a web server).