Disk space refers to the storage required to hold your website. Every single letter on your website has to be stored somewhere on your Web host's server. Each letter is one byte, and a megabyte is roughly 1 million bytes. HTML pages are usually very small, but images, scripts and databases can take up large amounts of disk space.
Bandwidth/transfer is the amount of data the server will allow your site to transmit in a month. Transfer is usually measured in gigabytes, or billions of bytes.