Bandwidth, which is also referred to as transfer, is the amount of data your web host allows your site to receive and send in a certain time period (usually a month).
Each character that your site receives (eg. a request for a webpage) of sends (eg. the requested webpage) counts as one byte. Most webpages are between 10,000-50,000 bytes in size. Extra media like images can easily range up to 150,000 bytes in size, and fine downloads (programs, mods, documents) can easily reach 10,000,000+ bytes (10 megabytes). Since files can be so large, and quite a few people can visit your website, most hosts list bandwidth in terms of gigabytes (billions of bytes).
If your site exceeds your monthly allocated bandwidth, most hosts will charge you an overage - an extra charge for all the extra bandwidth you used. This is usually done on dollars per gigabyte rate.
Please note that colocation hosting usually uses a different methodology of bandwidth pricing.
Bandwidth itself is not created equally. Imagine bandwidth a highway, and the data you transfer cars that pass through the highway. Now imagine a highway with 5 lanes versus a highway with only 2 lanes. So while in one hour we may be able to pass 1000 cars, the highway with 5 lanes will usually be faster. The same analogy applies online. Different carriers (providers of the internet connection) provide different levels of service. This can result in different download rates (bytes/sec - imagine the numbers of cars you can squeeze in the highway in one second) and latency (ping time - imagine how long it takes a car to get from Point A to Point B on the highway). Using cheap web hosting might result in low bandwidth, few features, and slow site performance.
Also to remember is that internet connections can, and will break. To that end, most web hosts have multiple carriers - there is more than one connection to your server. If one of the connections goes down, another one shoulders the load until it comes back up. It is prudent to investigate the providers a web hosting company is using. Also note that some web hosts designate carriers as backup connections. These connections are only used if the primary connection goes offline.