A continuous probability distribution of a random variable whose logarithm is normally distributed.
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable $X$ is log-normally distributed, then $Y=\ln(X)$ has a normal distribution. Equivalently, if $Y$ has a normal distribution, then the exponential function of $Y$, $X=\exp(Y)$, has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. (Wikipedia)
It is used for modeling financial returns and underlies the Black-Scholes-Merton model, among other. As every model, it is imperfect; logarithmic financial returns tend to have heavier tails than implied by the model.