Bandwidth is the amount of data that is allowed to be transferred in and out of an account during any given period (usually 1 month). For instance, if a client has a 1MB picture stored on their server and that picture is accessed 4 times, then 4MB of bandwidth has been used.
Bandwidth can be calculated in many different ways, but the majority of the time it is simple HTTP requests (accessing your website in the browser) that uses most of your bandwidth.