Only four parameters are needed to define everything that affects audio quality: Noise, frequency response, distortion, and time-based errors. Let's look at each of these parameters in turn.
Noise is the background hiss you hear when you turn your receiver way up, and you can also hear it during quiet passages when playing open reel or cassette tapes. A close cousin is dynamic range, which defines the span (expressed in decibels) between the background noise and the loudest level possible before the onset of gross distortion. CDs and DVDs have a very large dynamic range, so any noise you may hear was either from the original analog tape, was added as a byproduct during production, or was present in the room and picked up by the microphones when the recording was first made.
Subsets of noise are AC power-related hum and buzz, electronic crackling, vinyl record clicks and pops, between-station radio noises, tape modulation noise, cross-talk, windows that rattle and buzz at high volume levels, and the triboelectric cable effect. You're unlikely to notice tape modulation noise outside of a recording studio because it's specific to analog tape recorders, which are fast becoming obsolete, and usually hidden by the music itself. You can sometimes hear it if you listen carefully to a recording of a bass solo, where each note is accompanied by a "pfft" sound that disappears between the notes. The triboelectric effect is also called "handling noise" because it occurs when handling poorly made cables. I haven't seen a cable with this defect in about 20 years.
Frequency response is how uniformly a device responds over a range of frequencies. Errors are heard as too much or too little bass, midrange, or treble. For most people, the audible range extends from about 25 Hz at the low end, to just shy of 20 KHz at the high end. Even though many audiophiles believe it's important for audio equipment to respond to frequencies far beyond 20 KHz, in truth there is no need to reproduce ultrasonic content because nobody can hear it. Subsets of frequency response are physical microphonics, electronic ringing and oscillation, and acoustic ringing. These subsets are not necessary for consumers to understand, but they are important to design engineers and acousticians.
Distortion is the common word for the more technical term nonlinearity, and it adds new frequency components that were not present in the original source. When music passes through a device that adds distortion, new frequencies are created that may or may not be pleasing to the ear. The design goal for audio equipment is that all distortion be so low in level it can't be heard. I'll return later to the notion that distortion can be pleasing when I explain why some audiophiles prefer vinyl records and tube-based electronics.
There are two basic types of distortion - harmonic and intermodulation - and both are almost always present together. Harmonic distortion adds new frequencies that are musically related to the source. In layman terms, harmonic distortion adds a slightly thick or buzzy quality to music. All musical instruments create tones having harmonics, so a device whose distortion adds a little more merely changes the instrument's character by some amount. Electric guitar players use harmonic distortion - often lots of it - to turn a guitar's inherent plink-plink sound into a singing tone having great power and sustain.
Intermodulation (IM) distortion requires two or more frequencies to be present, and it's far more damaging because it creates new content that is not musically related to the original. Even in relatively small amounts, intermodulation distortion adds a dissonant quality that is unpleasant to hear. Another type of distortion is called aliasing, and it's unique to digital recording. Like IM distortion, aliasing creates new frequencies not harmonically related to the original, and so is unpleasant and irritating to hear. Fortunately, in all modern digital gear, aliasing is so low in level that it's inaudible.
Time-based errors affect mainly pitch and tempo. If you've ever played an old LP record where the hole was not quite centered, you've heard the pitch rise and fall with each revolution. This is called wow. Analog tape recorders suffer from a different type of pitch instability called flutter. Unlike the slow pitch change of wow, flutter is much more rapid giving a warbling effect. Digital recorders have a unique type of timing deviation called jitter, but with all modern equipment, jitter is so much softer than the music that you'll never hear it. The last type of time-based error is phase shift, but it's benign even in relatively large amounts.
Room acoustics could be considered a fifth audio parameter, but it really isn't. Nearby room boundaries can create frequency response errors (called comb filtering) due to wave reflections combining in the air. Reflections can also create audible echoes and reverb, but these are time-based phenomenon that occur outside the equipment, so they don't warrant their own category either. Likewise, with power amplifiers, maximum output power is important. But that's not related to fidelity - it merely states how loudly the amplifier can play.
The above parameters encompass everything that affects audio fidelity. If a device has noise and distortion too low to hear, a response sufficient to capture the entire range of audible frequencies, and time-based errors small enough to be insignificant, then that device will be audibly transparent to music and other sound passing through it. However, clarity and stereo imaging are greatly affected by room acoustics; without question, the room you listen in has far more effect on sound quality than any of the audio components.