Nuclear waste is the material that nuclear fuel becomes after it is used in a reactor. It looks exactly like the fuel that was loaded into the reactor -- assemblies of metal rods enclosing stacked-up ceramic pellets. But since nuclear reactions have occurred, the contents are’t quite the same. Before producing power, the fuel was mostly Uranium (or Thorium), oxygen, and steel. Afterwards, many Uranium atoms have split into various isotopes of almost all of the transition metals on your periodic table of the elements.
The waste, sometimes called spent fuel, is dangerously radioactive, and remains so for thousands of years. When it first comes out of the reactor, it is so toxic that if you stood within a few meters of it while it was unshielded, you would receive a lethal radioactive dose within a few seconds and would die of acute radiation sickness [wikipedia] within a few days. Hence all the worry about it.
In practice, the spent fuel is never unshielded. It is kept underwater (water is an excellent shield) for a few years until the radiation decays to levels that can be shielded by concrete in large storage casks. The final disposal of this spent fuel is a hot topic, and is often an argument against the use of nuclear reactors. Options include deep geologic storage and recycling. The sun would consume it nicely if we could get into space, but since rockets are so unreliable, we can’t afford to risk atmospheric dispersal on lift-off.
Nuclear reactors are typically loaded with Uranium Oxide fuel, UO2. Neutrons are introduced to the system, and many of them are absorbed by uranium atoms, causing them to become unstable and split, or fission, into two smaller atoms known as fission products. Sometimes, the uranium absorbs a neutron and does not fission, but rather transforms to a heavier isotope of uranium, such as U-239. U-239 beta-decays to Np-239, which in turn beta-decays to Pu-239. The heavier nuclide may then absorb another neutron to become an even heavier element. These heavier atoms are known as transuranics. Nuclear waste, with regard to nuclear reactors, is the collection of nuclides left over after a reactor has extracted some energy out of nuclear fuel. Many of the isotopes are very radioactive for a very long time before they decay to stability. The radioactivity causes the spent nuclear fuel to continue emitting heat long after it has been removed from the reactor. A few of the radioactive isotopes in the mix of spent fuel are gaseous and need to be carefully contained so that they do not escape to the environment and cause radiation damage to living things. Other types of nuclear waste exist, such as low level waste from other applications. This discussion will focus on high-level waste (HLW), the spent nuclear fuel from nuclear power reactors.
Spent nuclear fuel composition varies depending on what was put into the reactor, how long the reactor operated, and how long the waste has been sitting out of the reactor. A typical US reactor's waste composition is laid out in table 1. Notice that most of the Uranium is still in the fuel when it leaves the reactor, even though its enrichment has fallen significantly. This Uranium can be used in advanced fast reactors as fuel and is a valuable energy source. The minor actinides, which include Neptunium, Americium, and Curium, are very long-lived nuclides that cause serious concern when it comes to storing them for more than 100,000 years. Fortunately, these are fissionable in fast reactors and can thus be used as fuel! This still would leave us with the fission products. The decay of each nuclide vs. time is shown in Figure 1.
If all the electricity use of the USA was distributed evenly among its population, and all of it came from nuclear power, then the amount of nuclear waste each person would generate per year would be 39.5 grams. That's the weight of 7 U. S. quarters of waste, per year! A detailed description of this result can be found here. If we got all our electricity from coal and natural gas, expect to have over 10,000 kilograms of CO2/yr attributed to each person, not to mention other poisonous emissions directly to the biosphere (based on EIA emissions data).
If you want raw numbers: in 2002, there were 47,023.40 metric tonnes of high-level waste in the USA. 105,793 GW-days of thermal energy has been produced by nuclear power plants throughout the years to create that waste. Also in 2002, operating reactors added 2,407.20 metric tonnes  (1 metric tonne = 1000 kg).
Currently, nuclear waste created in the US is stored underwater in spent fuel pools near nuclear power plants. Assuming the DOE eventually licenses the Yucca Mountain repository in Nevada, this waste will eventually be stored deep underground. Since Yucca Mountain is on the Nevada test site, and since the area is geologically stable, the location is suitable. However, the repository is designed to a certain capacity of nuclear waste. If it ever opens, it will fill quickly thanks to the build-up of waste throughout the last few decades and another repository will need to be constructed. However, there are ways around this.
As mentioned previously, nuclear waste is over 90% uranium. Thus, the spent fuel (waste) still contains 90% usable fuel! It can be chemically processed and placed in advanced fast reactors (which have not been deployed on any major scale yet) to close the fuel cycle. A closed fuel cycle means much less nuclear waste and much more energy extracted from the raw ore.
France and Japan currently recycle spent fuel, although they only recycle one time before disposal. The US had a recycling program that was shut down because it created Plutonium, which is arguably the easiest material with which to make a nuclear weapon. Were some plutonium diverted in the recycling process, a non-nuclear entity could be one step close to building a bomb. However, under programs such as the (now stalled) GNEP [wikipedia], proliferation-free waste recycling can exist.
See Also See our main recycling page for a more thorough discussion of recycling.
The longest living nuclides in nuclear waste are the ones that can be used as fuel: plutonium and the minor actinides. If these materials are burnt in fuel through recycling, nuclear waste would only remain radioactive for a few hundred years, as opposed to a few hundred thousand. This significantly reduces concerns with long-term storage.
We could switch from Uranium/Plutonium based fuel to Thorium/Uranium-based fuel. This would allow for recycling and breeding without creating any plutonium or minor actinides whatsoever. Fission products are still created, of course, and some of them are quite long-lived, but reducing the minor actinides is a benefit of Thorium. A thorough thorium page is available here.
Swing by our radiation page to learn more about radiation in general.