A supernova is a star which has erupted on a terrific scale. When we see one in a distant galaxy, it can outshine all the other millions of stars that the galaxy contains. Our own Sun is not massive enough to go supernova, but that does not mean we are safe from them. One could still go off next door, on a cosmic scale of distance.
While experts do not agree exactly, it seems that a supernova within perhaps 100-200 light years of the Earth would have severe effects. The blast of x-rays, gamma rays and other cosmic radiation that it would produce would be most damaging. It could strip the ozone layer that protects us from the Sun's ultraviolet light. Then solar radiation would kill off some species, while depriving the rest of food and speeding up their rate of genetic mutation.
There are so few stars within 200 light years of the Earth that this is not a short-term risk we need fuss about. There is one possible future supernova, HR8210, only about 150 light years off. But even its possible explosion is hundreds of millions of years in the future.
However, the age of the Earth makes it pretty certain that there have been nearby supernovae within its lifetime. Heavy metal traces in rocks associated with the mass extinction at the end of the Ordovician, 440 million years ago, have been linked to a possible supernova. This extinction killed most extant species and was the most severe we know of. And the amount of time available means that there may well be another nearby supernova to attack life on Earth before the swelling Sun boils the oceans.