Of the two, storage is easier to understand. Just like you, we have hard drives (except larger and probably more expensive) and we store the files that make up your site on them. Storage refers to how much space those files take up. Storage billing is measured in units called megabyte-months, which refers to one megabyte stored on our system for one month.
Bandwidth, on the other hand, refers to the amount of data that is sent out when people visit your site. Bandwidth is measured in units of gigabytes. If your page (including any graphics and such that may go with it) takes up one megabyte of space, then about a thousand people (1024 actually) would have to download the whole thing to get to a gigabyte of bandwidth, and that's what you'd be billed for.