There are 51 stars within 17 light years of the Earth (source). If one of these stars was to become a supernova, how would they effect the Earth?
I have read the Wikipedia article Near-Earth supernova, but what would the explanation be in more layman's terms? What would happen if one of the 51 stars above became a supernova?
None of those stars can go supernova, so the question is rather moot. If you look at the classifications, the most luminous is Sirius A (an A sequence star even) you can get an idea of its mass. If you look at your source page, and link to the explanation you see that A stars range from 1.4 to 2.1 stellar masses. In order to go supernova though, you need about nine solar masses. The nearest supernova candidate is IK Pegasi (HR 8210), located at a distance of 150 light-years (and that's a Type Ia). The closest Type II candidate that I can think of is Spica, which is 260 light years away. Although this list doesn't include Spica, and has the closest Type II as Betelgeuse, at 640 light years away.
In his book, Death From The Skies, Dr. Phil Plait (a professional astronomer, writer, lecturer, etc.) covers exactly what would happen to earth if a star nearby did go supernova, and it isn't pretty. Basically, it would strip us of our ozone layer and turn the upper layer of our atmosphere into a dirty brown smog layer, let in a lot of UV radiation that would devastate plankton and plant life, and be very bad for all life down on the surface. However, for a Type II, they need to be around 25 light years or closer to affect us. Dr Plait's writing style is very much geared for a layman reader, and explains things in a very conversational tone, so I highly suggest that you get the book and read it. It will tell you exactly what to expect, and also tell you exactly why it's nothing to worry about.
All that said, there are two things in his book that you may want to consider as more likely than a supernova. A GRB from Eta Carinae, or an asteroid impact. The latter we may actually have the technology to eliminate as a concern though, so we're not doing too badly!
