Bandwidth (data transfer) is usually measured in gigabytes. A gigabyte is roughly one billion bytes. Each letter on this page is one byte. Images can take up a large amount of kilobytes (thousands of bytes), and file downloads can be several megabytes (millions of bytes) in size.
You need to realistically think of how much bandwidth your site will need. Unless your site has a lot of downloads or a large advertising push, it will be unlikely that you will need many gigabytes of transfer - 5 gigabytes should be enough. Any more, and you will just be paying extra for something to use. On that note though, you should be careful and make sure you always do have some extra bandwidth - many hosts charge $2.00 and more for every gigabyte of transfer you use extra. That can add up very quickly
Like most things, bandwidth is not created equally. There are varying degrees of performance delivered by carriers (the actual companies that provide the internet connection). One company may provide expensive, but super-fast connections while another could provide slow but cheap bandwidth. It is good to test it yourself by visiting the web host's page multiple times throughout the day to see how fast their response loads up. A slow loading website could be a sign of a slow and over-loaded internet connection.
Also like most things, internet connections can break. To ensure that your site does not go down, most hosts use multiple carriers for their internet connection. It is a good thing to look up a host's network of connection (usually found on its own or in the about section) and see that they have multiple connections. It is also important to note carriers designated as 'backup connections.' These carriers are only used when the primary connections go down. So while a web host might have a large network of internet connections, that does not matter if only a couple are actually being used.
Lastly, the size of the pipe should be noticed. The 'pipe' is the actual internet connection, and its size refers to the speed at which data is transferred. Just like a home connection can be 33.6k and 56k (which refers to the speed of the connection), internet connections are measured in Gig-E, OC3, DS3, and so forth. The faster the pipe, the more data it can handle, and thus usually the better it is.