Supernova are the explosions of massive stars more than 8 times the mass of our sun. Their cores run out of fuel, and because nothing is pushing out and opposing their own gravitational pull, the stars just collapse in on themselves, forming either a neutron star or a black hole. As they go supernova, they produce a hugely powerful shock wave that basically explodes the star.
Mostly, astronomers just see the aftermath of supernovas because the estimated rate of supernova production in our galaxy is about one every 50 years, and the explosions happen in a very short timespan of only about 100 seconds. The remnants they leave behind remain bright for some time, and their peak brightness can last for months, so it depends what you mean by “seeing” a star go supernova—seeing the star in the initial stage of the explosion is very, very rare; we’re much more likely to see the remnants. Many well-known nebulae are supernova remnants, including the Crab Nebula:
But one of the first such remnants to be discovered soon after its star went supernova was SN 1987A, in the outskirts of the Tarantula Nebula in the Large Magellanic Cloud. The light from it first reached Earth on February 23, 1987. It was the first supernova to be observed across every band of the electromagnetic spectrum, and because it was so close, it could be seen with the naked eye. It was the first opportunity for astronomers to observe a supernova up close, and it gave them fascinating insight into the process of stellar evolution, allowing them to test theories against observations.
But the first time astronomers have witnessed a star go supernova in real-time was in 2008. NASA’s orbiting Swift telescope spotted the extremely luminous blast of X-rays of SN 2008D while watching another older supernova.
Astronomers were able to observe the evolution of the explosion right from the start, catching the initial X-ray signature of the stellar death of SN 2008D, which means they now know what X-ray patterns to look for, and can hopefully in the future spot more supernovae at the same critical moment.