While the other answers are not wrong, I don't think they adequately address the fundamental conceptual hurdle to understanding characteristic impedance.
Imagine you are a wave. You propagate by taking a step - these steps are always the same size. This is your wavelength.
The characteristic impedance is the impedance, or resistance, you will feel taking each step. Low impedance might feel like walking normally, while high impedance might feel like walking through mud - there is a lot more viscosity resisting the movement of your leg every time you take a step.
Now, the total energy or loss or however you'd like to look at it is going to very much depend on length, and it does. But it doesn't matter how far you have to go, it is going to be a certain difficulty taking one step through air, and one step through mud. The characteristic impedance is the impedance felt taking one step. The number of steps there are to take does not change this value.
To bring things back into reality a bit and less analogy, the characteristic impedance is the impedance one wave length of a propagating electromagnetic wave will 'feel' through a given transmission line. This is why it is called the characteristic impedance - it's an impedance that characterizes the bulk nature of it. At any given step, the signal is going to see the same impedance between it and the next step.
This is why one can terminate a 50 Ω transmission line with a 50 Ω resistor on one end regardless of length - one can view the termination as the final 'step' the wave makes on its transmission journey, so a lumped 50 Ω resistance across the transmission line pair is perfectly acceptable - because the wave has already been experiencing 50 Ω of impedance at all times.
Now, let's take this conceptual understanding as context, and touch on The Phonon's excellent answer.
Knowing that the characteristic impedance is, in fact, the impedance felt at any given time when traveling down a transmission line, it becomes obvious that this is also the ratio of voltage to current that won't cause a reflection.
However, this might still be confusing. Wouldn't that mean that higher frequencies, having to take more steps, would experience that much more resistance for the same length of line? Well, attenuation down a transmission line does generally increase with frequency, but not because of this.
Let's assume you get the 'characteristic' part of characteristic impedance. But, you also need to get the impedance part. Impedance is a complex value, meaning it has both real and imaginary components.
Imaginary in the mathematical sense - don't fall into the trap of taking imaginary in a mathematical context literally. It's a name, that's all.
Imaginary numbers are named as such as sort of a play on words compared to the name we gave the opposite basis number line - real numbers. All numbers are, technically, imaginary. Likewise, no numbers are real. But some are imaginary. And some are real.
Real numbers and imaginary numbers form the complex plane, which can be imagined as two axes at right angles, one being the real number line, stretching from -∞ to ∞, the other being the imaginary number line, stretching from -∞*i to ∞*i. And, we know they exist and we need them because there are equations whose solutions demand imaginary numbers. Without them, you simply ignore the ability to answer an entire category of equations. At the simplest, imaginary numbers allow us to give an answer to this equation: \$x^{2} + 1 = 0\$. x, of course, is equal to i.
OK, that was a bit of a tangent, but a valid understanding of complex numbers is absolutely required before one can understand impedance.
Impedance is made up of a real component, which is simply DC resistance, and an imaginary component called reactance. Reactance is apparent resistance, but it is not due to the dissipation of energy as heat (as with resistance), but rather the temporary storage of energy that is later released. If you see energy being siphoned off because it is getting stored in an electric field (aka a capacitor) or a magnetic field (inductor), at that moment, it appears just like energy that is simply lost as heat due to resistance.
It depends on the transmission line, but they of course do suffer increasing losses with length. You will usually find this somewhat indirectly given as 'attenuation per foot' or attenuation per 100 meters or similar, in dB/. This will include losses due to real resistance (which is not even as simple as measuring with an ohmmeter - frequency will change things like skin depth, making the same conductor appear more resistive, etc., etc.), dielectric loss, whatever other things cause a true dissipation of energy into entropy/heat.
Characteristic impedance is, generally, almost entirely due to reactance. So 50 Ω of reactance and 0 Ω of resistance wouldn't actually cause any loss - it would only be a temporary loss as energy is stored, but then later released, back into the line. If you have a voltage and current ratio that is not such that the voltage drop (energy stored) at a given current is equal to the voltage across the transmission line, then you do not perfectly balance the energy stored with the energy released, and you get the bane of signal integrity's existence, REFLECTIONS!! Oh noes!
This cycle of energy storage and release forms a standing wave in our transmission line. Any excess voltage forces more current to flow, which means we've exceeded the energy storage capacity of the cable, so the phase gets thrown off, and our standing wave gets destructively interfered with. Our signal is, to varying degrees, destroyed.