Half-Life of Radioactive Isotopes: Why? How?
As commenters have pointed out, there is nothing about half-life that makes it special; we could as easily measure and tabulate one-tenth lives or one-fifth-lives. 

Half-life has two marginal advantages: the half that has decomposed and the half that hasn't are equal, so whether we are referring to remaining or already decomposed nuclei is unambiguous, while for a third-life we could be confused as to whether we are talking about the time it takes for a third of the nuclei to decompose or for all but a third of the nuclei to decompose; and experimentally speaking, half is convenient because it's distinctly smaller from the initial amount of nuclei but large enough to be easily measured compared to the initial amount. However, these two advantages are fairly unimportant, and if we had decided to measure and tabulate one-tenth lives for all radioactive isotopes, we would be fine.

What's conceptually more important is that we can use half-lives as a proxy for isotopic disintegration rates, because radioactive decay follows first-order kinetics. Nothing prevents us from using half-life to describe any kinetic process, such as a chemical reaction rate. However, most of the time that's of limited value, because in general, half-lives are not constant, but change depending on the amount of reagents. It is only for first-order kinetics (i.e. when the amount of nuclei decays exponentially with time) that half-lives (or any fractional life) are constant.

For a general kinetic equation with only one reagent $\ce{A}$,

$$ -\frac{\mathrm{d} [\ce{A}]}{\mathrm{d} t} = k [\ce{A}]^n$$

where $n$ is the kinetic order of the process, we can always define the half-life of the process by integrating from an initial amount of $\ce{A}$, $N_\ce{A}$, to half that amount:

$$ \int\limits_{N_\ce{A}}^{N_\ce{A}/2} \frac{\mathrm{d} [\ce{A}]}{[\ce{A}]^n} = \int\limits_{0}^{t_{1/2}} - k\,\mathrm{d} t $$

which means that the half-life can be expressed in terms of the integral on the left side:

$$ t_{1/2} = - \frac{1}{k} \int\limits_{N_\ce{A}}^{N_\ce{A}/2} \frac{\mathrm{d} [\ce{A}]}{[\ce{A}]^n}  $$

This integral will in turn depend on the kinetic order $n$. While in general the half-life will take the form

$$ t_{1/2} = \frac{1}{k} \frac{1}{n-1} \left( (N_\ce{A}/2)^{1-n} - N_\ce{A}^{1-n} \right) $$

and therefore will be dependent on the starting conditions $N_\ce{A}$, and will change as $\ce{A}$ is consumed, for first-order kintetics ($n=1$)

$$ t_{1/2} = - \frac{1}{k} \left( \ln{|N_\ce{A}/2|} - \ln{|N_\ce{A}|} \right) = \frac{\ln{2}}{k} $$

so half-life does not change with the amount of $\ce{A}$.

As long as the decay is first-order (or exponential, which is the same), half-life is such a constant that we can measure and tabulate, as it will be the same under all possible circumstances; no matter if we are talking about tonnes of fissile material or a handful of nuclei, the value for half-life will be the same.

If we start with $N_\ce{A}$ radioactive nuclei, after a time $t_{1/2}$ you'll have half that amount, $N_\ce{A}/2$; since now you have $N_\ce{A}/2$, after another $t_{1/2}$ you'll have $N_\ce{A}/4$; after a third interval equal to $t_{1/2}$ you'll have $N_\ce{A}/8$, and so on.

This makes tabulating half-lives equivalent to tabulating kinetic coefficients $k$, as these can be transformed into each other quickly and unambiguously, while for other kintetic orders we'd need additional informations to perform that transformation.

Note as well that the last step also shows why we could arbitrarily decide to use any other fractional lives in the same way: if, for instance, we used one-tenth lives, we'd get

$$ t_{1/10} = \frac{\ln{10}}{k} $$