divided by total area under diagonal.
What's a Lorenz curve? A lorenz curve is a graphical plot of the percentage of total income (or of whatever else is being measured; could be wealth or consumption) against percent of the unit of analysis, typically population, families, or households. Thus if 50 percent of the population had 50 percent of total income, the data point would be 50,50, hence 'on the diagonal.' If income was divided absolutely evenly, x percent of people would always have x percent of income, hence the plot would be a diagonal. Insofar as income (or whatever) is distributed unevenly, the lorenz curve bends down and away from a diagonal 45-degree line.
Dividing one area by another does away with any unit of scale, leaving a simple number that is always less than one. the more area between the curve and the diagonal, the more inequality, and the higher the gini. A single, summary measure is useful in comparing countries across time and/or space.
There are a number of different ways to measure inequality. Gini is the most popular. One problem is that lorenz curves for two different situations could cross, leaving a comparison indeterminate. Another is that other methods can give you different results. There is a cottage industry of very arcane research on this that I was never interested in. Simpler is better, IMO, which means considering the trends in median whatever, or the shares of whatever in terms of percentiles.
A good, comprehensible reference is The Economics of Inequality by A.B. Atkinson.