Hello,
Transmission via cable with specified characteristic impedance serves several goals.
1. RF Power amplifiers want to see a specified load to operate properly
2. Low noise amplifiers also want to see a specified source impedance to give optimum noise performance.
3. High speed signals can be affected by reflection when the back and forth traveling time is no longer small w.r.t signal duration (1/Baudrate).
When wires between a source and a load are no long short with respect to Wavelength, strange things start to appear due to reflections.
A piece of coaxial cable that is electrically 0.25 lambda long appears as an almost short circuit when the end is left open (refer to quarter wave transformer).
When you have a load resistor of 50 Ohms and connect this via a 75 Ohms cable to a source, this source may experience every impedance between 50 and 112 Ohms.
To avoid reflection (think of analog video), the 75 Ohms cable has to be terminated at least on one side. Mostly 2 sides are terminated as even small reflection may lead to shadow images.
Reflection in digital systems result in signal distortion that reduces the noise margin. The actual value is not of importance, but certain realization techniques are limited to certain values.
A power amplifier that wants to see a 50 Ohms load, mostly doesn't have an output impedance of 50 Ohms. Compare this with your audio amplifier at home, It wants to see (for example) 4 or 8 ohms, but its output impedance is generally far below these values.
Besides 50 and 75 Ohms coaxial cable, you also see 100..120 Ohms (twisted wire pairs), 92 Ohms coaxial 62 Ohms coaxial, 300 Ohms symmetrical, 600 Ohm symmetrical (HF feeder to the antenna. Probe leads for an oscilloscope have the highest impedance (very thin inner conductor) to reduce capacitance.