Random Link: A capacitor is, strictly speaking, a source of distortion. It introduces lag (charge/discharge) and partly depresses a waveform's natural shape on charge.
"Capacious" distortion in Izotope Trash 2, the shape's exaggerated, but it portrays audible capacitor distortion nicely.
ESR is a type of distortion mostly cheap electrolytic capacitors have. ESR means "equivalent series resistance". Capacitors are also resistive to an extent (like, yes, resistors), at higher frequencies their impedance grows. The worse a capacitor's quality, the worse is ESR: some very cheap and ugly capacitors can be used as lowpass filters right away, at around 3-5 KHz (no guarantee about quality or exact frequency, mind it). This can be tried (and listened for) with headphones - simply solder some low-value caps in series with voice coils. Resistors and wires are also capacitive to an extent, but that's a topic for another article.
ESR is a lot better in modern good-quality aluminium or tantalium electrolytic capacitors than it used to be in, say, 1970s. It's almost non-existent in the better parts. Still, it's one reason why electrolytic capacitors are inferior to other types.
The superior types are teflon, polystyrene, and silver mica. All those types simply distort less, there is less lag (and time-stretch) between charge and discharge and they tend to preserve shape better. Typical capacitor waveform distortion, by the way, is around 5%. Some are better than others (again, teflon, polystyrene and silver mica). Teflon caps are expensive and difficult to get, but polystyrene capacitors are very common and popular (Sprague "Orange Drop" series capacitors are a popular sort for guitar upgrades). Silver mica caps can be bought off online stores and sourced in Russia.
Here it must be said that the elusive warmth in electronic music playback is simply waveform accuracy/fidelity. Frequency and dynamic response matter here. As an example, guitar players (and serious hi-fi fellows) prefer valve amplifiers simply because their dynamic response in treble and high midrange is better than transistor amps'. Valve amplifiers have much better (lower) Miller capacitance than transistor amplifiers (in plain words, they lag less in the higher frequencies). The same applies to capacitors: the better a capacitor's dynamic response (the less waveform lag/time-stretch), the livelier/warmer its output is.
Capacitors are used as simple tone lowpass circuits in guitars. A potentiometer and a capacitor are used to cut off high frequencies. This capacitor directly affects dynamics (the purpose of a tone capacitor bypass switch is to get rid of this slight, but noticeable slowdown). A low-quality tantalium capacitor is usually slow/stiff, dull in output; a polypropylene capacitor is fast and lively. This is not as noticeable with clean output as with effects (overdrive/chorus/delay). A guitar's own woolliness also affects output - e. g. a basswood glued-neck guitar will always be slower and duller than a maple neck-through guitar. Some fanatics insist there is no audible difference between guitar tone caps, but the reason why is this - they don't play with enough effects or a quick/sharp enough guitar.
Capacitors can also be used to correct distortion - e. g. in some headphones/speakers, which tend to overshoot too much on incoming current (usually underdampened, either electrically or physically or both). It's as simple as sum of a capacitor's negative phase response and a device's overly positive phase response.