Complex Plane
No, not this complex plane ...  
... this complex plane: 
It is a plane for complex numbers! 
(It is also called an "Argand Diagram")
Real and Imaginary make Complex
A Complex Number is a combination of a Real Number and an Imaginary Number.
Let me explain ...
A Real Number is the type of number you are used to dealing with every day.
Examples: 12.38, ½, 0, 2000
With real numbers we can do things like squaring (multiply a number by itself):
2 × 2 = 4
But what can we square to get 4 (minus 4)?
? × ? = 4
Well, 2 won't work because multiplying negatives gives a positive: (2) × (2) = +4, and no other Real Number works either.
So it seems that mathematics is incomplete ...
... but we can fill the gap by imagining there is a number that, when multiplied by itself, gives 1 (call it i for imaginary):
i^{2} = 1
Now, we can do this:
2i × 2i = 4i^{2} = 4 × (1) = 4
And together:
A Complex Number is a combination of a Real Number and an Imaginary Number
Examples: 3.6 + 4i, 0.02 + 1.2i, 25  0.3i, 0 + 2i
The Complex Plane
You may be familiar with the number line:
But where do we put a complex number like 3+4i ?
Let's have the real number line going leftright as usual, and an imaginary number line going updown:
And we can plot a complex number like 3 + 4i
:
It is placed


And here is 4  2i : It is placed

And that is the complex plane:
 complex because it is a combination of real and imaginary,
 plane because it is like a geometric plane (2 dimensional).
Whole New World
Now we can bring the idea of a plane (Cartesian coordinates, Polar coordinates, Vectors and more) to complex numbers, and we open up a whole new world of numbers that are more complete and elegant, as you will see.
Complex Number as a Vector
You can think of a complex number as being a vector.
This is a vector.
It has magnitude (length) and direction.
And here is the complex number 3 + 4i
as a Vector: 
You can add complex numbers as vectors, too:
Here we add the complex numbers 3 + 5i and 4 − 3i as vectors: Add the real numbers, add the imaginary numbers, like this: (3 + 5i) + (4 − 3i) = 3 + 4 + (5 − 3)i = 7 + 2i 
Polar Form
Again, here is the complex number 3 + 4i
As a Vector: 

Here it is again (still as a vector), but In polar form: 
So the complex number 3 + 4i can also be shown as distance (5) and angle (0.927 radians).
How do you do the conversions?
Example: the number 3 + 4i
We can do a Cartesian to Polar conversion:
 r = √(x^{2} + y^{2}) = √(3^{2} + 4^{2}) = √25 = 5
 θ = tan^{1} (y/x) = tan^{1} (4/3) = 0.927 (to 3 decimals)
We can also take Polar coordinates and convert them to Cartesian coordinates:
 x = r × cos( θ ) = 5 × cos( 0.927 ) = 5 × 0.6002... = 3 (close enough)
 y = r × sin( θ ) = 5 × sin( 0.927 ) = 5 × 0.7998... = 4 (close enough)
In fact, a common way to write a complex number in Polar form is
x + iy = r cos θ + i r sin θ = r(cos θ + i sin θ)
And "cos θ + i sin θ" gets shortened to "cis θ"
So 3 + 4i = 5 cis 0.927
Summary
The complex plane is a plane with:
 real numbers running leftright and
 imaginary numbers running updown.
To convert from Cartesian to Polar Form:
 r = √(x^{2} + y^{2})
 θ = tan^{1} ( y / x )
To convert from Polar to Cartesian Form:
 x = r × cos( θ )
 y = r × sin( θ )
Next ... learn about Complex Number Multiplication.