Here’s the mathematician’s definition (From the Open University):

a plane making use of Cartesian coordinates in which the x-axis represents the real part of a complex number and the y-axis represents the imaginary part.

Here’s the history (cut-and-pasted from http://www-groups.dcs.st-andrews.ac.uk/history/Mathematicians/Argand.html):

The first to publish this geometrical interpretation of complex numbers was Caspar Wessel. The idea appears in Wessel's work in 1787 but it was not published until Wessel submitted a paper to a meeting of the Royal Danish Academy on 10 March 1797. The paper was published in 1799 but not noticed by the mathematical community. Wessel's paper was rediscovered in 1895 when Juel draw attention to it and, in the same year, Sophus Lie republished Wessel's paper.

This is not as surprising as it might seem at first glance since Wessel was a surveyor. However, Jean Robert Argand (1768-1822) was not a professional mathematician either, so when he published his geometrical interpretation of complex numbers in 1806 it was in a book which he published privately at his own expense. His knowledge of the book trade allowed him to put out this small edition but one would have expected it to be in a less noticable place than Wessel's work which after all was published by the Royal Danish Academy. Perhaps even more surprisingly, Argand's name did not even appear on the book so it was impossible to identify the author.

And, finally, here is a more accessible (I hope) explanation.

It is a way of representing complex numbers in a visual and geometric way. And also, a very powerful tool to help mathematicians and engineers describe real-world behaviour of physical systems.

At school, we learn about the number line. Then, for a while, we learn about geometries: Cartesian coordinates (x, and y), polar coordinates (r and theta), and so on.

The next big step is to put these things together, and realise that the whole 2-dimensional space is a number field (as opposed to a number line). One way to describe a point in this field is with the Descartes’ x- and y-coordinates, but another, much more powerful, way is to use these strange things called imaginary numbers. Along the horizontal axis we have the conventional number line, made up of real numbers 1, 2, 3… and -1, -2, -3…. Up the vertical axis, we have a new number line, made from the imaginary numbers i, 2i, 3i…. And of course, -i, -2i, -3i. The geeks call it an Argand diagram, but it is really only an x-y graph with a different name.

Now we learn that any place in the number space can be described in terms of these so-called complex numbers: -5 +4i. It’s a number, and it behaves just like the positive real integers, but it describes a place in space. Cool! (well, if you enjoy math, it’s cooler than Eddie).

The next step is to look at other ways of describing this point in space. Think of angles and lengths from the origin. If the point in space is labelled 1+ i, (that’s 1, 1 in Cartesian coordinates) it is at an angle of 45 degrees (or pi/4 in radians, which are only cool to mathematicians), and has a length of 1.4142… (2^0.5). What is the square of this complex number? Double the angle, square the length, and you get 2i.

How about the cube? Three times the angle, cube the length, and you get 2i-2.

This works for all imaginary numbers, and complex ones, What’s the cube root of –1? Anything of unit length which gets you to 180 degrees (pi radians) when you triple its angle, so that’s pi/3 and –1 (pi) and 5pi/3 (make that 60 degrees, 180 degrees and 300 degrees, or (3^0.5+i)/2, -1, or (3^0.5-i)/2.

From there you can do some really neat tricks with these imaginary numbers, plotting pretty well any curve you like in 2-Dimensional space.