A complex number is a number of the form $~$z = a + b\textrm{i}$~$, where $~$\textrm{i}$~$ is the imaginary unit defined as $~$\textrm{i}=\sqrt{-1}$~$.
Doesn't make sense? Let's backtrack a little.
Motivation - expanding the definition of numbers
Say we start with a child's understanding of numbers as counting numbers, or natural numbers. Using only natural numbers we can add and multiply whichever numbers we want, but there are some restrictions on subtraction: ask a child to subtract 5 from 3 and they'll say "you can't do that!".
In order to allow the computation of numbers like $~$5-3$~$, we need to define a new kind of number: negative numbers. The set of natural numbers, negative numbers and $~$0$~$ is called whole numbers, or integers. Unlike natural numbers, with integers we can add and subtract whatever we want. (In math terms, we say the set of integers is closed to addition.)
We still have a problem, though: what about division? Going back to our child analogy, say we're talking to a third grader. If you ask them to divide 8 by 2, no problem. But what if you as them to divide 2 by 8? "You can't do that!"
Just like with negative numbers, we need to define a new kind of number: fractions, like $~$\frac{1}{2}, \frac{5}{3}$~$ or $~$-\frac{6}{7}$~$. By adding fractions to the integers, we get a more general set of numbers - rational numbers. With rational numbers we can multiply and divide however we want - the set of rational numbers is closed to multiplication. Well, except dividing by zero. No one has figured out how to do that without breaking mathematics, so that's awkward.
Now say our child is a seventh grader, so they know about square roots. They can apply the square root operation to any perfect square, no problem - $~$\sqrt{9}=3$~$. But what if you ask them to find the square root of a non-perfect square, like $~$\sqrt{2}$~$? "You can't do that!"
Once again, we need to expand our set of numbers to include a new kind of number: irrational numbers. Unlike we discussed with negative numbers or fractions, irrationals can do a whole lot more that just define square roots. In fact, irrationals are so cool that according to some sources the Pythagoreans couldn't deal with their existence and ended up killing the guy who invented them.
Adding irrational numbers to our set of rational numbers, we get the real numbers. Mathematically, we say that the reals are a [real_number_completeness complete] ordered field, which is actually one of the ways of defining them.
However, there's still one problem: the real numbers are not an [-algebraically_closed_field]. In a practical sense, this means we still don't have closure under the square root operation $~$\sqrt{}$~$, because we can't define the square roots of negative numbers.
By now you've probably noticed the pattern, though: any time someone says "you can't do that!", mathematicians invent a new kind of number to prove them wrong.
Introducing: imaginary numbers
In order to allow using the square root operation on negative numbers, we once again have to define a new kind of number: imaginary numbers.
Actually, we have to define just one number - the imaginary unit, $~$\textrm{i}$~$. $~$\textrm{i}$~$ is defined as a solution to the quadratic equation $~$x^2+1=0$~$. In other words, we can define $~$\textrm{i}$~$ as equaling $~$\sqrt{-1}$~$.
All the other imaginary numbers (square roots of negatives) follow directly from this definition of $~$\textrm{i}$~$: for any negative number $~$-a$~$, we can define $~$\sqrt{-a}=\textrm{i}\sqrt{a}$~$.
To be continued…
This article is unfinished, and will later include an explanation of the complex plane as well as the algebraic properties of complex numbers.
[todo: Finish article by adding explanations of the complex plane and the algebraic properties of complex numbers.]