Different type of capacitors

How can I distinguish between filter capacitors, decoupling capacitors, and bypass capacitors? What is the most basic and important difference in their functionality?

How to distinguish between the filter capacitor, decoupling capacitor, a bypass capacitor?

All capacitors have a fundamental mathematical formula associated with their operation and, that formula applies to all uses of the capacitor i.e. a capacitor can be used in a filter or as a decoupler or, in an integrator or in whatever circuit that needs to use a capacitor BUT, it will act the same way embodying the same mathematical formula.

Hence, the distinguishing feature is defined by the circuit schematic rather than the actual part/model of the capacitor.

In other words, there is no such thing as a "filter" capacitor but, there is such a thing as a circuit that uses a capacitor to realize a filter function.

what is the most basic and important difference in their functionality.

There is no difference; they all utilize this formula: $$\i = C\frac{dv}{dt}\$$.

Of course, some capacitor types are better suited to some filter applications and, some other capacitor types might be better suited to different filter applications. This picture might help you understand: -

Image taken from here.

Bypass capacitors and decoupling capacitors are the same thing - that's simply two names for the same function.

Here's an example circuit diagram:

simulate this circuit – Schematic created using CircuitLab

In that diagram, C1 is a filter capacitor. It is used with R1 to make a simple low pass filter (cutoff of ~1060Hz.)

C2 is a bypass capacitor, also known as a decoupling capacitor. It is used to keep noise on the power supply from getting to the op-amp. It is also used to keep noise from an IC from getting into the power supply.

Both capacitors have the same value, and in reality may even be identical parts - a simple 100nF ceramic disk capacitor could be used in both places.

The names given to them (filter capacitor, decoupling capacitor) come from their function, not from the way they are made.

There will be a tendency to use certain capacitor types for certain functions.

Ceramic capacitors are often used for decoupling because their very low equivalent series resistance makes them well suited for handling the (relatively) large currents involved.

Signal filters often use film capacitors instead of ceramic capacitors because ceramic capacitors are known to be "microphonic" - mechanical vibration can cause changes in capacitance that will cause changes in the signal. Tapping a PCB can cause clicks and thumps in the audio signal passing through a ceramic capacitor. Film capacitors on the other hand may have a somewhat higher equivalent series resistance that makes them less suited for use as a decoupling capacitor.

Some are polarized, like electrolytic one while others are not. i believe the smallest electrolytic capacitor is 0.47uF, 470nF.
470nF is already a pretty big capacitor in value.
an electrolytic capacitor is simply 2 metal plates wrapped around with a insulator in between.

they are pretty "big" in psychical size.

a bypass capacitors usually 100nF, a common agreed standard between engineers, used to filter out noise from the power supply or prevent noise generated by the device it's used on to spread on the power rails.

Film capacitors from Wima are film capacitors, the type of insulation inside can differ, the small one is 5mm x 2.5mm, the big one is 7mm x 3.5/4.5/5 mm.

some are insulated with: https://en.wikipedia.org/wiki/Polypropylene, they are on the expensive side, but good for Audio applications.

22pF, pico farad capacitors are usually mounted across an opamp to filter imperfections of opamp that could create considerable noise. they are small, the metal plates inside are insulated with ceramic material.

1 pF (pico farad capacitor) can be made in SMD construction. inside are alternating metal plates with insulation in between the plates, the format is 0603, Length: 1.55 ± 0.05 mm Width: 0.85 ± 0.05 mm Height: 0.45 ± 0.05 mm, 0201 SMD package measures 0.024" x 0.012" which is 0.6 x 0.3 mm.

the size of the capacitor is determined by the value. pretty hard to create a electrolytic capacitor of 1, 10 or 22pF. the cost would be astronomical to buy one.