It's all about transmission lines, and transmision lines are all about energy transfer. While it may seem new and exciting to you, transmission lines have been around for a long time. Why are we cursed with the plethora of cables today? Because electronics is an exercise in compromise.
Leafing through the 1945 edition of The radio amateur's handbook we read "A transmission line is used to transfer power, with a minimum of loss, from the transmitter to the antenna..."(1) Herein lies a key difference in the way we look at transmission lines. We are not interested in the power transferred from a sinusoidal generator. Although the transmission line behaves no differently between the two cases, we use some different phrases.
We also see the characteristic impedance defined as "The characteristic impedance of a transmission line, also known at the surge impedance, is defined as that impedance which a long line would present to an electrical impulse induced in the line."(1) For parallel conductors in air the approximation is z=(276/(k**0.5))*log(D/a) where a is the conductor radius and D the center to center spacing of the conductors. For a coaxial line z=(138/(k**0.5))*log(b/a) where a is the outside diameter of the center conductor and b is the inside diameter of the outer conductor. k is the dielectric constant of the separating medium.
Where is the compromise? We have resistive losses in the wire and losses in the dielectric. At some voltage there will be a breakdown across the dielectric from the shield to the center conductor. We need to choose between power transfer, attenuation and voltage breakdown. Although the optimal physical configurations are close to each other, they are not the same. Power transfer is optimal in a coaxial cable with a 30 ohm characteristic impedance. Attenuation is optimized, for the minimal amount, at about a 73 ohm characteristic impedance. The maximum performance if you're worried about voltage breakdown is at about 60 ohms.
Why do we use 50 ohm cables? It has to do with the size of commercially available copper tubing and common wire sizes during the development of RF technology. No special reason.
The voltage standing wave ratio is a measure of the impedance mismatch between two elements along a transmission line. It refers to the ratio of the current at a node to the current at an antinode. This assumes a sinusoidal wave generator at the source. We do not have a sine wave generator as our source, but this is still meaningful to us.
"...the input impedance of a transmission line terminated in its characteristic impedance is simply equal to the characteristic impedance, independent of the length of the line. For this reason ... a transmission line is most often terminated in its characteristic impedance."(2)
Terminating a transmission line in other than its charateristic impedance leads to difficulties. "That is, the incident wave is reflected at the output of the line because of the improper termination at the receiving end. Reflections are avoided in pulse applications particularly since pulses that are reflected back and forth between the input and output ends of the line completely obscure the true input signal."(2)