Capacitors are components used in an electronic circuits that have multiple functions including storing charge, regulating voltage, filtering frequencies, coupling and decoupling, creating oscillating circuits and protecting circuits. In the very simplest sense they store and release energy like a battery, though this energy is utilised in many more different ways.
In electric pianos they are found in many circuits, from a single film capacitor on the bass boost potentiometer on a Fender Rhodes thats acts as a variable high pass filter, to over two dozen inside the Wurlitzer piano amplifier where their function varies widely.
The energy stored in capacitors is called capacitance, and is measured in farads. Usually in electronics this is in smaller units such as microfarads (uF) and picofarads (pF). A picofarad is 1,000,000 times smaller than a microfarad so 1uF = 1,000,000pF.
This article will discuss the general use of capacitors in electric pianos, which includes film capacitors. There are several different types of capacitor. See the relevant links for more details:
See Electrolytic Capacitor
Here is an explanation of some of the functions of capacitors in electric pianos.
Coupling and Decoupling:
One of a capacitor’s main functions is to allow AC signals (such as an audio signal) to pass through them while blocking DC signals. In an amplifier, the AC signal is the audio signal from the instrument, and the DC signals provide the power and biasing to control the electronic components to increase this signal.
A capacitor that is placed in series in a circuit can block the DC current from one area entering another. This means that different parts of the amp that require different voltages are not affected by each other. A capacitor placed in this way is known as a coupling capacitor. It is essential for the correct operation of an amp as different sections have very different voltage requirements. For example the preamp and power amp stages of a Wurlitzer 200A amp are separated by a capacitor (C8) in series.

A capacitor in a decoupling role has a different purpose: primarily to stabilise the voltage and reduce noise from the DC current in the circuit. When a capacitor is placed in parallel in a circuit with one end going to ground; it provides a low impedance path for unwanted noises and signals. A good example of this is the high voltage section the Wurlitzer 200 series amplifiers, where three 0.33uF capacitors filter out noise as to not interfere with the electrostatic reed bar functions.

Smoothing Fluctuations:
When the power to a circuit is enabled, capacitors charge up then store energy. This energy can then be dissipated in the circuit if there is a ripple or evenness in the DC supply. This has the effect of smoothing out the current which leads to a more consistent operation and less noise. Capacitors can be particularly useful for this when placed in parallel shortly after diodes that convert AC voltage to DC voltage. Because smoothing this current this can require a high capacitance, electrolytic capacitors are often used. They can store much high capacitance for their size. For example in a Wurlitzer 200A amplifier, C28 and C29 are two 2200uF electrolytic capacitors in the power amp section.

Filtering:
Capacitors alongside resistors and inductors can filter certain frequencies, creating low-pass, high-pass, and other EQ filters for the audio signal. In a Hohner Clavinet, the switches “Soft, Medium, Treble, Brilliant” that shape the EQ of the signal are controlled in this way. The circuit here is a series of high pass filters that are bypassed when the switch is engaged. This high pass filter effect happens because when capacitors are placed in parallel they filter the lower frequencies to ground due to the higher impedance of the capacitor, removing the low frequencies from the signal.

Vibrato
In a Wurlitzer amplifter, there is a circuit dedicated to vibrato. This is achieved by creating a low frequency oscillator (LFO) which then manipulates the signal. Capacitors are central to this process because the oscillations are creating by capacitors quickly charging and discharging. This is controlled by resistors in a circuit creating a resistor-capacitance filter.
The more capacitance a capacitor has, the slower the oscillation of the LFO will be, and vice-versa. In a Wurlitzer amplifier, the capacitors in the vibrato circuit create an LFO which triggers an LED and light dependent resistor (LDR) component. This LED/LDR is what changes the amplitude of the audio signal, causing the tremolo sound. An LDR is used after the capacitor because it is smoother, and creates the classic warm vibrato tone. Nonetheless, the capacitor is at the heart of this process – setting the rate of the tremolo.

Ceramic Capacitors
Many of the capacitors here have a high capacitance range, but smaller ceramic capacitors are used in electric piano circuits to eliminate noise and other high frequencies. This is achieved similar to the effect of decoupling: a ceramic capacitor with a very low capacitance (often lower that 500 picofarads), is inserted in parallel in the circuit. This is often the case around transistors, the transistor can amplify all of the signal, including high frequencies. As high frequencies have low impedance in capacitors in parallel circuits, a ceramic capacitor here creates a path for the unwanted high frequencies to go to ground, eliminating them from the signal.

Safety
Capacitors, especially in amplifiers like the Rhodes Suitcase and Wurlitzer pianos, can remain charged quite a while after the power is off. Always discharge the capacitors before removing or working on them, especially capacitors with higher capacitance such as electrolytic caps and in circuits with higher voltages and currents.
Note: It may be helpful to view the full schematic of the references if you want to learn more or get a better idea of how the capacitors fully work. Below are the links to them: