How is xΩ impedance cable defined?
I see you have some accrurate but probably difficult to understand answers.  I'll try to give you a better intuitive feel.

Consider what happens when you first apply a voltage to the end of a long cable.  The cable has some capacitance, so it will draw some current.  If that were all there was to it, you get a big current spike, then nothing.

However, it also has some series inductance.  You can approximate it with a little series inductance, followed by a little capacitance to ground, followed by another series inductance, etc.  Each one of these inductors and capacitors models a little length of the cable.  If you make that length smaller, the inductance and the capacitance goes down and there are more of them in the same length.  However, the ratio of the inductance to the capacitance stays the same.

Now imagine your initial applied voltage propagating down the cable.  Each step of the way, it charges up a little capacitance.  But, this charging up is slowed down by the inductances.  The net result is that the voltage you applied to the end of the cable propagates slower than the speed of light, and it charges the capacitance along the length of the cable in a way to require a constant current.  If you had applied twice the voltage, the capacitors would get charged to twice that voltage, therefore would require twice the charge, which would take twice the current to supply.  What you have is the current that the cable draws being proportional to the voltage you applied.  Gee, that's what a resistor does.

Therefore, while the signal is propagating down the cable, the cable looks resistive to the source.  This resistance is only a function of the parallel capacitance and series inductance of the cable, and has nothing to do with what it connected to the other end.  This is the characteristic impedance of the cable.

If you have a coil of cable on your bench that is short enough so that you can ignore the DC resistance of the conductors, then this all works as described until the signal propagates to the end of the cable and back.  Until then, it looks like a infinite cable to whatever is driving it.  In fact, it looks like a resistor at the characteristic impedance.  If the cable is short enough and you short the end, for example, then eventually your signal source will see the short.  But, at least for the time it takes the signal to propagate to the end of the cable and back, it will look like the characteristic impedance.

Now imagine that I put a resistor of the characteristic impedance accross the other end of the cable.  Now the input end of the cable will look like a resistor forever.  This is called terminating the cable, and has the nice property of making the impedance be consistant over time and preventing the signal from reflecting when it gets to the end of the cable.  After all, to the end of the cable another length of cable would look the same as a resistor at the characteristic impedance.