We say that a random variable X has a lognormal distribution if its logarithm, Y = log X, is normally distributed. The normal distribution often occurs when a random variable comes about by combining a bunch of small independent contributions, but those contributions combine additively; when the combination is multiplicative instead, lognormals occur. For example, lognormal distributions often occur in models of financial markets.
But of course X = exp Y, so the variable we care about is the exponential of a normal. Why isn't it called expnormal?