The number of nuclei left after time $t$ in radioactive decay is given by:
$$N(t) = N_0 e^{-t/ \tau}$$
Now if we put $N(t)$ as $\dfrac{N_0}2$, we get half-life. But, if we had put $\dfrac{N_0}4$, we would have quarter-life, which is also independent of $N_0$.
Is there anything special about half-life as opposed to quarter-life
The decay time $t_{1/2}$ of half the given number $N_0$ of atoms atoms is just convenient and visually appealing. Of the unit fractions it is also nearest to the decay time constant (mean lifetime) $\tau$ $t_{1/2}=0.6931 \tau$. The decay time to a unit fraction $1/n$ given by the positive integer $n$ is $$t_{1/n}=\tau \cdot ln(n)$$
Radioactive decay is an example of an exponential random process. Two key statistics for any exponential random process are the median and the mean. (The standard deviation is equal to the mean for an exponential random process.)
The half life $t_{1/2}$ is the median. The time constant $\tau$ is the mean. There's nothing particularly meaningful about the quarter life.
