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 many different ways, but the majority of the time it is simple HTTP requests (accessing your website in the browser) that use up most of your bandwidth.
Bandwidth is always stated as a monthly limit in either MB or GB. This is the amount of actual data that can be transferred from the web server that holds your website, to the browser of the person who is looking at your website. If you go over the limit, then either you will have to be an extra fee, or your website will not be available. The amount of bandwidth you use is obviously related to the amount of traffic (visitors) that you receive.
There are web hosting companies that offer Unlimited bandwidth. This sounds like a good deal. The truth is that this is nothing but a lie and you should steer clear of web hosting companies that offer unlimited bandwidth. Bandwidth is a commodity that costs the web hosting company money - unlimited bandwidth is not only impossible, but would cost an unlimited amount of money. When web hosts offer unlimited bandwidth they are trying to trick you. They know that you will probably not use very much bandwidth. The moment you try to use a lot of bandwidth - then you will find your website shutdown.
For most web sites you will not need much bandwidth. Probably 99% of all websites use less than 2 GB of bandwidth a month. If you intend to have a lot of downloads of software, audio or video, then you may end up using a lot more bandwidth, maybe 50 GB. Adult sites typically use a lot of bandwidth. Sites like mp3.com require an enormous amount of bandwidth, but for the average site - a low level will be enough in almost all cases.