What does this equation mean?

The property that makes radioactive decay useful as a clock is not that it happens on a convenient timescale — some isotopes decay in fractions of a second, others over timescales longer than the universe’s age — but that it happens with no memory. An atom of uranium-238 that has existed, undecayed, for four billion years is exactly as likely to decay in the next second as one formed a microsecond ago. There is no wear, no fatigue, no accumulated probability of failure the way there is in a mechanical clock’s escapement or a living organism’s senescence. This absence of memory is what makes decay first-order: the number of decays in a short interval is proportional only to the number of atoms present, not to their age, giving the population as a whole an exponential decline, $N(t) = N_0\, e^{-\lambda t}, \qquad t_{1/2} = \frac{\ln 2}{\lambda}$. where $N_0$ is the initial population, $\lambda$ is the decay constant characteristic of the isotope, and $t_{1/2}$ is the half-life derived from it. This is the one place in the article where the evolutionary-population analogy is exact rather than suggestive: a sample of $N_0$ identical atoms behaves, statistically, exactly as a population geneticist’s model of a large population with a fixed per-individual probability of an event per unit time behaves. Individual atoms are unpredictable; the population’s decline is not.