Bandwidth, sometimes just called "speed", can be a confusing topic.
Internet providers usually claim a certain bandwidth for a given service. This tells how fast data can travel through the connection. In principle, digital bandwidth is measured by how many bits (digital units of data) can travel across the connection in a given amount of time. Usually the time interval is one second. But since many thousands, or these days, even many millions, of bits can cross a connection in a second, bandwidth is usually measured in thousands of bits per second (kbps or kb/s) or these days even in millions of bits per second (Mbps or Mb/s).
This is further complicated by the fact that often the download bandwidth (the speed at which bits travel from the Internet to your computer) is often different from the upload bandwidth (the speed at which bits travel from your computer to the Internet. Most users are much more concerned with download bandwidth, but increasingly upload bandwidth is becoming important for people who create large volumes of data (e.g. music or video) or who use file sharing.
Often people write bandwith as download bandwidth followed by a slash and then upload bandwidth. For example one might write RCN's standard bandwith (at the time of this writing) which is 5 Mbps download and 384 kbps upload as 5 Mbps/384 kbps. To keep things simple on this site, I have tried to write bandwidth always in Mbps, so I'd write that as 5 Mbps/.384 Mbps, or 5M/.384M or even 5/.384 for short. Sometimes people leave off the units and are not consistent, in which case you need to have a general idea of what's available to know what they are talking about. For example if one were to write RCN's standard bandwidth as 5/384, you'd have to guess that the 5 was in Mbps and the 384 was in kbps. By standardizing on Mbps, I hope to avoid some of this confusion on this site.
This is a hard question. More is always better, but how much it's worth to you is a personal decision. It depends on what you use your Internet connection for, and how snappy you like your Internet experience to be.
For a lot of people who do typical web browsing and email and not much else with the Internet, download bandwidth that's at least about .2Mbps or so and upload bandwidth that's at least .05Mbps should be adequate. That means that pretty much any DSL or cable modem connection should do it. Nevertheless, even with just that kind of usage, you will probably notice faster download speeds up to a few Mbps. After a few Mbps you probably won't notice any further increases. Whether this difference matters to you is a matter of preference. The best thing to do is try the Internet connections of some of your friends and ask them the bandwith (many won't know!)
People who work with large amounts of data will benefit more from even higher bandwidth. The problem is that these days it's harder and harder to know how much data you are using. Web pages themselves tend to be pretty small, but the graphics files in them are sometimes pretty big (in the hundreds of kb) so if you want to download those pages in a second you'll want your download bandwidth to be at least in the hundreds of kbps (.1 Mbps or better). But the real bandwidth hogs for most people are music and video. If you just download, only the download bandwidth really matters, but if you upload (e.g. if you email media files or use file sharing) you need to consider your upload bandwidth also.
I can't go into any detail on this topic because there are so many reasons, but here are a few of the most popular: