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?
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!
Just to add to the answer above, scientists can make assumptions as to how Earth would be affected by a nearby supernova, particularly by one of the main effects, as mentioned in Larian's answer - a gamma ray burst.
According to the article "Did a gamma-ray burst initiate the late Ordovician mass extinction?" (Melott et al. 2004), a GRB is a probable contributor to the mass extinction that occurred around 440 million years ago. The 2 possible mechanisms for the mass extinction discussed by the authors are:
Increased UV radiation reaching the surface, due to the depletion of the ozone layer
the increased production of nitrogen dioxide haze that would cause a global cooling (as was observed to also be a contributing factor in the Ordovician extinction).