A gamma-ray burst (GRB) is a brief flash of gamma rays coming from an astrophysical source at great distances from us, often from hundreds of millions of light years away. Gamma rays are a kind of light (like visible light, microwaves, or X-rays) that is very energetic, and whatever produces gamma rays must therefore contain (and unleash) a large amount of energy in a very short amount of time. Thus the study of gamma ray bursts is a study of some of the most violent events in the universe.
GRBs were discovered in the late 1960s and early 1970s by Earth-orbiting satellites designed to keep watch against covert nuclear weapons testing, but it was quickly realized they originate well outside our solar system. Their origins remained mysterious for several decades because they came and went so quickly -- often within a few seconds -- and because their position couldn't be pinpointed to better than a few degrees on the sky. For comparison, the full Moon is about 0.5 degrees across, and a telescope at moderate magnification has a field of view less than half that -- a few degrees is a huge area to search, especially when you've only got a few seconds to do it!
In 1997, an Italian-Dutch satellite known as Beppo-SAX used its combination of gamma-ray and x-ray detectors to pin down the precise position of a gamma-ray burst, and it was soon discovered that the source of the burst was at cosmological distance -- hundreds of millions of light years away. In the decade since this discovery, hundreds of GRBs have been localized, some having both X-ray and optical counterparts, and all of them have been associated with explosive events in galaxies at very large distances from us.
There are two kinds of gamma-ray burst, known as long-soft and short-hard, referring to their duration, and the nature of their gamma-ray emission. Long-soft bursts last for a few dozens of seconds, and emit less energetic ("soft") gamma rays; short-hard bursts last for a second or less and emit very energetic ("hard") gamma rays.
The long-soft GRBs are the ones which have been detected most often at other wavelengths, and they are believed to be associated with the collapse of supermassive stars, in an event known as a hypernova. When a massive star runs out of the nuclear fuel that makes it shine, the core of the star collapses. If the core collapses into a black hole, the remainder of the star will begin to fall onto it. Black holes sometimes produce jets of material that fly away from the black hole at close of the speed of light, and in a hypernova, the infalling stellar material acts as a source for these jets. These events probably happen dozens of times a day across the entire universe, but we only detect them as a gamma ray burst if, by chance, the jet from the black hole happens to be pointed in our direction. GRBs produce the most intense radiation along the direction of the jet, and so we only detect them when they're pointed right at us.
Although they haven't been studied as well, the short-hard GRBs are also believed to originate from the formation of a black hole. In this case astrophysicists think they come from the merger of two black holes or two neutron stars in orbit around one another. Both black holes and neutron stars are very massive and very, very small in size, and when they orbit one another closely, they move very fast! If they spiral together and merge with one another, their collision may result in a huge explosion that occurs very quickly, producing a very rapid burst of gamma rays at high energies.
Most of the energy emitted by a gamma-ray burst comes out as gamma-rays, but the jets that create them and the resulting hypernova emits light at other wavelengths too, and by studying the afterglow, you can learn more about the object that created the GRB than you can from just studying the gamma ray emission. The light emitted in X-rays, optical light, and radio waves can often persist for hours or days after the gamma ray burst, and because of the nature of radiation at these wavelengths, it is easier to pinpoint where the GRB is from the afterglow than it is from the gamma ray burst itself. You can also figure out what kind of star it was that exploded, how the explosion progressed, or what the environment was like around that star by studying the afterglow.
GRB afterglows are hard to find, but there is now a network of space satellites and ground-based observatories dedicated to their detection and localization. Satellites like Swift are designed to quickly detect and localize GRBs to much higher precision than was previously possible. Satellites can now provide gamma ray localizations to less than 0.5 degrees (sometimes much less), making it easier for ground-based observers to concentrate their search on a particular spot on the sky. The satellite radios the coordinates back to Earth, and these coordinates are then relayed to observatories around the world via the Gamma Ray Burst Coordinates Network or GCN (of which the AAVSO International High Energy Network is a part). Observers can then turn their telescopes toward those coordinates, and search for a transient -- an object not previously observed at those coordinates. If they find one, then it's possible that they're looking at the GRB afterglow. The discoverer of an afterglow usually communicates the exact position and their initial observations to the rest of the GCN community, and other observers around the world begin observing the object, too. If the object fades in brightness over the next few hours or days and doesn't move as a minor planet, comet, or asteroid in our Solar System would, then they've found the afterglow! Continued observation of the source gives information on how the GRB explosion evolves, what its environment is like, and sometimes even what the progenitor object was.
We think we understand the basics of how GRBs happen, but we don't know everything, and sometimes we see some surprising things when we study gamma ray bursts and their afterglows. Sometimes the gamma ray light curve is very complex, with lots of rapid changes, sometimes not; sometimes the GRB afterglow light curve seems to evolve like a supernova, sometimes not; sometimes two GRBs with very similar gamma ray light curves will have totally different light curves in the optical, or perhaps one might not have an optical afterglow at all. We still don't completely understand what happens during a GRB, and the more observational data we have the better our understanding will be. Often, we learn more by encountering something we don't expect to see than by seeing what we expect.
There are many observatories searching for afterglows, including many robotic telescopes that search for them automatically, so your chances of being the first to observe an afterglow is not as great now as it was in the first decade of the 21st Century. But robotic telescopes can't be everywhere at once, and there is still a chance for individuals to make contributions to GRB research, especially by obtaining photometry during the outburst.