Great thread this. Full of useful information. There are some very knowledgable people contibuting to the thread.
But I think we risk getting ourselves into a confusion. IMHO you've got to draw a distinction between waste created by uranium mining and reactor waste.
In uranium mining you're dealing with a low level radioactive waste, in quite large volumes of tailings (which usually stays at the minesite). It's radioactive firstly because of the residual unextracted uranium (most of the uranium >90% finishes up in Japan or Europe (or China?) and secondly because of the residual naturally-occuring uranium series nuclides that were left over after the uranium is extracted. The half-life of U-238 being 4.5 billion years, it hangs around for a long time - in fact just as long as it would if it hadn't been mined. The tailings are usually disposed of on or near the minesite.
When 'yellowcake' is produced from the uranium ore at the processing plant, usually near the minesite, it contains most of the uranium from the ore and not too much of the daughters in the uranium decay chain. (Just as an aside, would you believe me if I told you that 'yellowcake' isn't yellow at all? It's actually a very dark green would you believe, almost black.)
The yellowcake is shipped overseas to be processed into fuel rods. The processing is for two reasons - firstly to get the uranium into a form that can be fabricated and machined into the rod and secondly so that the useless (non-fissionable) U-238 can be reduced and the useful (fissionable) U-235 can be increased. This latter processed is called enrichment - the most common isotope of uranium - U238 -is removed (and subsequently used for ammunition, for yacht keels and for elevator counter-weights). The U-235 (which in its natural ore-form is about 0.7% abundant) is enriched to a suitable amount to match the reqirements of the reactor that it is going to fuel. [The enrichment process is invariably carried out with the uranium in a gaseous form - uranium hexafluoride]
So when a fuel rod is loaded into a reactor, it's radioactivity is all due to U-238 and U-235. These are two long-lived isotopes and the rods are not very radioactive at all.
When the reactor is operated for a while, you get neutrons smashing into uranium atoms. When a neutron hits U-235, it breaks the atom into a couple of pieces (i.e. the fissioning event) and also produces more neutrons which then smash some more U-235 etc (i.e. the chain reaction). The two pieces broken from the U-235 atom are, in fact, lighter atoms such as strontium or iodine and they are likely to be radoactive. They are the so-called fission products, and they decay with their own half-lives.
But what is happening to the U-238 while all this fissioning of U-235 is going on, I hear you ask. Well, the U-238 (probably more abundant than U-235) doesn't deign to fission (i.e. split) because the neutrons are the wrong energy. What it can do is absorb a neutron into its nucleus and turn into a heavier element called plutonium. When the reactor burnup is complete and the fuel rod taken out for replacement, it contains all the (non-gaseous) fission products and the plutonium that has been bred.
And this material is broadly called reactor waste and is the subject of much research on number of schemes designed to dispose of it safely
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