Sampling, reproduction, and transmission distortions
On wire type
When you consider a thing you want to measure, there is such a thing as sampling too slowly.
Consider wheels on television (and assume that your 25 or 30 frames per second are samples much faster than 1/25 or 1/30 seconds, so that blurring is not an issue here).
Spokes on these wheels are spaced at regular intervals, some angle apart, so if the wheel turns exactly that angle between two video frame samples, the next image will look identical, and the wheel will appear not to move while the car does. If you're slightly slower or slightly faster, you will see frames that suggest it is turning very slowly in one way or the other (at some speed, and often direction, that clearly doesn't match the car's movement). Due to the practicalities of wheel size and car speed, spokes often turns multiples faster than the video capturing/reproduction, which gives even more arbitrary frames when the wheel changes speed, which is why it more easily appears to switch direction.
Which is a long winded analogy to point out the inability to measure something happening faster than your sampling speed.
If you try, you will still happily store things, but represent and reproduce something that never happened.
...at a lower frequency than that it did in reality. Let's shift to sound. Consider a 1000Hz sine wave and sampling it at exactly 1000 samples per second. You will catch the wave at exactly the same point in its oscillation, which will yield a flat line.
Nyquist states that a frequency of half the sample rate is the highest you're going to find in your recording (for example, 1000 samples can at best express the high and low points of a 500Hz wave). Put another way, the Nyquist frequency is the frequency whose period is two sampling intervals.
Also, this is just for the one frequency, and the real problems are the speeds around the Nyquist frequency. Higher ones will appear as lower-frequency non-harmonic content; consider:
This mainly implies that you should filter out frequencies above the Nyquist frequency before sampling. That is, you wish to limit the signal bandwidth to that of your recording device, and there are some practical aspects to this. Theoretically, you could sample at a frequency just larger than the signal bandwidth, but this requires an a filter to counteract the aliasing that would perfectly passes frequencies up to this point and perfectly remove frequencies above it. Such a filter cannot be realized, so in reality some degree of oversampling (or some filtering out of the signal) is necessary.
For example, 44KHz recording tends to apply a filter that cuts out above 20KHz - commonly a lowpass filter that has its transition band between 20 and 22kHz. (Note that since we barely hear 20Hz anyway, any distortion around there matters less)
Things at and just under the Nyquist frequency won't be perfect, though only in amplitude, and this is rarely a problem. Consider a sine wave; you'll catch two points from the sine wave, hopefully a high and a low point, as they are half the wavelength apart, but so are the zero crossings, so you'll get something between zero and the correct amplitude. This effect also applies somewhat for frequencies just below as well.
Note that there is suggestion that the wheel analogy mis-suggests: Frequencies just under the Nyquist frequency will not alias, because of the way Fourier Transform works, specifically because it bases frequency content on a fairly large window of samples.
In a wider sense, aliasing refers to any distortions that come from forgetting the above effect. While 44kHz is more than enough for a straight sample-then-reproduce process, this is not the end of the story.
Various transformation, such as resampling, pitch effects, time/frequency-domain conversions, and digital effects may implicitly change the frequency content to introduce those beyond the Nyquist frequency. These details mean you may need additional lowpass filtering to avoid aliasing.
Quantization refers to an effect of digital sampling, that of taking analogue values and recording it as discrete numbers.
Quantization error refers to the difference between the analogue values and the digital values.
The lower the resolution is, the higher that error is. Take a fairly pathologically bad example case like 3-bit audio. There are 8 levels in total, which would make your digitized wave very blocky (...depending a little on what you decide that data represents). Since real sound tends to be sinusoidal, the difference between the smooth original and the blocky levels that heavy quantisation introduces would be very audible.
An interesting detail is that that difference (=quantization error) is related to the original signal, which can work out to be distracting.
An Analog-to-Digital Converter (ADC) is a quantizer by nature, so quantization error always applies. Early sound cards were 8-bit. It was decided fairly quickly that this is not enough (not even for reproduction). More recently 16-bit digital audio is very common, which is enough for production (from masters at good levels).
More is useful for recording (mostly for the practical problems of 'how do I set my volume level'), and also editing and transformation (in part for the practical reason of re-quantization/roundoff errors in this intermediate work). Basically, it helps to start with more detail than what you want to end up with.
Rumble refers to vibrations from turntable mechanics (motor, bearing, belts, pulleys) being introduced into the signal.
This is impossible to avoid completely, but can be minimized.
Wow and flutter refers to effects from the variation in (analog) mechanically driven reproduction, such as in tapes.
In digital systems, a similar variation-in-timing effect still applies, but on a much smaller timescale, and is referred to as jitter.