Complex Impedances
TL;DR The imaginary part of the impedence tells you the reactive component of the impedance; this is responsible (among others) for the difference in phase between current and voltage and the reactive power used by the circuit.
The underlying principle is that any periodic signal can be treated as the sum of (sometimes) infinite sinewaves called harmonics, with equally spaced frequencies. Each of them can be treated separately, as a signal of its own.
For these signals you use a representation that is like: $$ v(t) = V_{0} \cos (2 \pi f t + \phi) = \Re \{ V_{0}e^{j 2 \pi f t + \phi} \} $$
And you can see that we already jumped in the domain of complex numbers, because you can use a complex exponential to represent rotation.
So impedance can be active (resistance) or reactive (reactance); while the first one by definition doesn't affect the phase of signals (\$ \phi \$) the reactance does, so using complex numbers is possible to evaluate the variation in the phase that is introduced by the reactance.
So you obtain: $$ V = I \cdot Z = I \cdot |Z| \cdot e^{j \theta} $$
where |Z| is the magnitude of the impedance, given by: $$|Z|=\sqrt{R^2+X^2}$$
and theta is the phase introduced by the impedance, and is given by: $$\theta = \arctan \left( \frac{X}{R} \right) $$
When applied to the previous function, it becomes: $$ v(t) = \Re \{ I_{0}|Z|e^{j 2 \pi f t + \phi + \theta } \} = I_{0} |Z| \cos (2 \pi f t + \phi + \theta ) $$
Let's consider the ideal capacitor: it's impedance will be \$ \frac{1}{j \omega C} = -\frac{j}{\omega C} \$ which is imaginary and negative; if you put it into the trigonometric circumference, you obtain a phase of -90°, which means that with a purely capacitive load the voltage will be 90° behind the current.
So why?
Let's say that you want to sum two impedances, 100 Ohm and 50+i50 Ohm (or, without complex numbers, \$ 70.7 \angle 45 ^\circ \$ ). Then with complex numbers you sum the real and imaginary part and obtain 150+i50 Ohm.
Without using complex numbers, the thing is quite more complicated, as you can either use cosines and sines (but it's the same of using complex numbers then) or get into a mess of magnitudes and phases. It's up to you :).
Theory
Some additional notions, trying to address your questions:
- The harmonics representation of signals is usually addressed by Fourier series decomposition:
$$ v(t) = \sum_{- \infty}^{+ \infty} c_{n}e^{jnt} , \text{ where } c_{n} = \frac{1}{2 \pi } \int_{-\pi}^{\pi} v(t)e^{-jnt} \, dt $$
- The complex exponential is related to the cosine also by the Euler's formula:
$$ cos(x) = \frac{e^{ix}+e^{-ix}}{2} $$
I am sure this will not answer entirely your question, in fact I hope this will complement the answers already given that seem to neglect: the concept behind the use of complex numbers (which, as already said, is just a fancy name for a type of mathematical "quantity", if you will).
The first main question here we should answer is why the complex numbers. And to answer this question we need to understand the need of the different sets of numbers, from the natural until the real numbers.
From the early ages the natural numbers allowed people to count, e.g, apples and oranges in a market. Then the integer numbers were introduced to address the "in debt" concept by means of negative numbers (this was a hard concept to understand at that time). Now, things get more interesting with the rational numbers and the need to represent "quantities" with fractions. The interesting about this numbers is that we need two integers, and not only one (as with the natural and integer numbers), for instance 3/8. This way of representing "quantities" is very useful, for instance to describe the number of slices (3) left in an 8 slices pie, when 5 were already eaten :) (you could not do this with an integer!).
Now, let us jump the irrational and the real numbers and go to the complex numbers. Electronics engineers faced the challenge of describing and operating a different type of "quantity", the sinusoidal voltage (and current) in a linear circuit (i.e, made of resistors, capacitors and inductors). Guess what, they found that complex numbers were the solution.
Engineers knew that sinusoids were represented by 3 components, that is, A (amplitude), \$\omega\$ (angular frequency), and phase (\$\phi\$): $$y(t) = A \cdot sin(\omega t + \phi)$$
They also realized that in a linear circuit the angular frequency (\$\omega\$) would not change from node to node, that is, no matter which point in the circuit you were probing, you would only see differences in terms of amplitude and phase, not frequency. They then concluded that the interesting (varying) part of a sinusoidal voltage (or current) was its amplitude and phase. So, just as we do with the rational numbers we need two numbers to represent the varying sinusoidal voltage in a linear circuit node, in this case (A, phi). In fact they realized that complex numbers algebra, that is, the way you operate and relate these numbers to each other fits like a glove with the way sinusoids are operated by linear circuits.
So when you say that the impedance of a capacitor is \$ \frac{1}{j \omega C} \$ i.e, (A=1/C, phi=-90º) in the above adopted notation, you are actually saying that the voltage is delayed 90º regarding the current phase. And please, forget that "transcendental" nomenclature about imaginary and complex... in fact we are talking about "quantities" with two orthogonal components (i.e, "that don't get mixed no matter how hard you shake them in a cocktail cup"), just like vectors, that represent two different physical aspects of the phenomena.
UPDATE
There are also some notes I highly recommend to read, "An Introduction to Complex Analysis for Engineers" by Michael D. Alder. This is a very friendly approach to the subject. In particular, I recommend the first chapter.
Using complex numbers is a mathematical way of representing both in phase and out of phase components - the current with respect to the voltage. Imaginary impedance doesn't mean that the impedance doesn't exist, it means that the current and voltage are out of phase with each other. Similarly a real impedance doesn't mean real in the everyday sense, just that the current is in phase with the voltage.