The half-life is due to spontaneous decay. They're kind of unstable so there is a chance of them just going "poof" and decaying. Half-life is the time it takes for half of the atoms in any given amount of material to undergo this spontaneous decay.

However, in a reactor we've arranged it so that there is a pretty big chance that when one atom decays the neutron (small sub-atomic particle) shoots out and hits another atom, which will cause that atom to split and shoot off more neutrons, which will hit other particles and cause a cascade effect. Compare it to just shooting a billiard ball on a pool table randomly vs stacking the balls into a pyramid (shooting a ball into that pyramid will cause a whole bunch of other balls to move around).

That effect is used in a reactor, because when a whole bunch of little atoms decay quickly they release heat. In our normal powerplants they're stacked so that it happens very quickly (although our current generation is kind of inefficient and only a small percentage of the fuel is used up before the effect slows down or becomes too difficult to handle due to dangerous byproducts), and that generates a lot of heat which is used to heat water into steam and drive a steam turbine.

There is also something called a radioisotope thermoelectric generator, a nuclear battery of sorts. Basically a radioactive element that has been arranged to encourage it to just react a little bit faster. Not cascade, just generate enough heat that it can be used to generate power through the thermoelectric effect. Those batteries last a lot longer, so they're used on deep space satellites like Voyager (that travel far enough away from the sun that solar panels aren't useful anymore). Voyagers nuclear battery produced something like 60% of its original power back in 2001 (some 25 years after its launch) but theoretically a battery like this could be designed to last thousands of years.