Depleted Uranium is uranium metal from which the % of Naturally Occuring U-235 has been lowered from about 0.7% to about 0.2% or less. U-235 is fissile and is required in high percentages for uses in nuclear weapons and nuclear power. (Though fuel rods only require about 5% U-235 while weapons grade material requires upwards of 85% pure U-235). When natural uranium is refined to produce the fissile U-235, it leaves behind a 'waste' product of the non-fissile U-238. Because U-238 cannot go critical and is simply an alpha emitter with relatively weak gamma rays, the physical properties of the metal can be used. Depleted Uranium is incredibly cheap, but incredibly dense and pyrophoric when in a fine state. Because of the high density (~20 g/cc) and the very cheap price, it makes great use as a counterweight where high mass but low volume is needed. (Tungsten could also be used, but it's MUCH more expensive than DU is).
Depleted Uranium is also VERY good at blocking radiation. Yes, it seems kind of odd that a radioactive metal would be good at blocking radiation, but with a very long half-life and relatively weak alpha and gamma emission, the radiation it blocks is far more damaging than the radiation it gives off. Therefore, many high intensity radioisotopes used in research or for medical uses are housed in depleted uranium containers.
Because of the high density and easy machinability, DU has found used in munitions for guns. The incredibly low cost, incredible hardness, and propensity to explode if it breaks apart on impact makes it a very good tool for piercing the armor of tanks and heavy machinery. In the same way, DU covered in lead and then steel makes very good, very dense, very inexpensive, and very effective armor plating.
Outside of the body, DU is virtually harmless. The alpha particles won't get through the layer of dead cells in your skin, and the gamma rays are pretty weak. It's not something you'd want to have lying out in the open so you could continuously expose yourself to it, but it's not going to severely increase your radiation exposure. Inside the body, however, DU is both chemically and radiologically toxic. That's because the uranium's alpha particles are now able to get at freshly created cells and cause internal damage. Plus, the DU accumulates in your kidneys where it causes major renal damage both chemically and radiologically. Therefore, inhaling the fumes of burning DU is very dangerous, and you don't want any of it getting inside of you. (This is why there's the big uproar about using DU in military missions. Sure it's effective at what it does, but all the burning DU and vaporized uranium is easily inhaled by the soldiers and the people living in that area).
One MAJOR reason why DU is carefully watched, however, is because if U-238 absorbs a moderately slow neutron, it will easily be convereted into U-239 which then decays via beta decay to Np-239. Np-239 has a half-life of about 2.5 days afterwhich it becomes the HIGHLY fissile Pu-239. So if someone had a source of neutrons and a LOT of time, they could easily create some plutonium. Thankfully, it's not as easy as putting neutrons into your DU. You need a good deal of money to safely do the conversions, and any other materials you might need to do this nasty stuff would raise some flags. (Like if someone went to buy a few pounds of DU as well as some beryllium or aluminum. It would raise some red flags. In fact, just trying to acquire large amounts of DU would raise some red flags since it is also used as the tampering device in a nuclear bomb).