The term "amplification" in acoustic context is ambiguous and often misleading. While it is widely used and accepted, it's not correct: you can use it, but with awareness.
You are having correct doubts about aspects related to electric amplification, and, interestingly enough, you are involuntarily getting to the point: it doesn't really matter if we're talking about sound or electricity.
Sound is still energy.
To extend what correctly pointed out in other answers, since energy is always conserved (in some form), it cannot be created from nothing, but must be transferred ("added") in some way, so that the original vibrations can be amplified.
In order to amplify something, you always need to add energy from somewhere else.
Electric (real) amplification
Let's take the example of voice amplification (instead of stringed instruments with sound boards, I'll explain why later).
A microphone is an electrical device that transforms kinetic energy of a membrane (that moves following the vibrations in the air) into changes in an electric current: when there's no air pressure, there is no passage of current, when the pressure increases, more current passes through it. Since sound is based on variations of pressure (including negative pressure) over small amounts of time, you get lots of changes in the current.
This is an oversimplification, but try to follow me anyway.
Then that current comes and returns through a powered circuit that is able to multiply (amplify) it. For instance, if it gets current 1, it becomes 4, if it gets 2 it becomes 8, if it gets 0, it remains 0.
Finally, that current goes to a speaker, that works like an inverted microphone, transforming electric signals into kinetic energy: more current, more pressure. And the resulting sound is much louder than the original.
Before going on on the concept of "natural amplification", let's understand what actually happens whenever we hear a sound.
As we all know, physical sound is an abstract concept that, for us humans, is the result of variations of air pressure in time. We need to have frequent changes in air pressure in order to perceive sound, and we need something that alters those pressure changes.
When some object is in contact to the air and that object starts to "vibrate", changes in its physical displacement create pressure waves that may eventually arrive to our ears.
Pressure changes expand radially in waves from their origin point (the classic pebble in the pond example), and eventually dissipate after some time, possibly transforming the kinetic energy into thermal energy in the meantime.
If the waves encounter a different pressure (like a harder surface), they may be reflected back in the opposite direction, meaning that a certain point may receive back a wave similar to one they already got.
But if the waves encounter a surface that is able to absorb ("stop") that kinetic energy, those waves are easily dissipated. That energy doesn't disappear, it is normally transformed in small amounts of heat.
An electric guitar has a fixed body that absorbs most of the vibrations of the strings, meaning that most of the potential energy that could eventually be transmitted from those strings is practically nullified. The pick-up is able to transform a small part of the energy of the vibration (causing changes in the magnetic field) into an faint but sufficient electric signal, usually between 100 and 300 mV.
Acoustic ("natural") amplification
Simply put: it doesn't exist. That's because, in normal conditions, no energy is being added. If you want "more sound", it means that you want "more energy", but, as said above, you cannot create energy from nothing.
So, how is it possible that, with the same sound source, we can get "louder" sounds?
The reason is that "acoustic amplifiers" are, in reality, energy transformers and better energy "conductors". They are more efficient in transforming and propagating the original vibrations (including those of physical objects like strings).
When you get a louder sound, you're not getting more energy than originally produced, but more of the originally produced energy, which is otherwise dissipated in other ways as explained above (heat).
The acoustic megaphone
An acoustic megaphone just redirects the acoustic energy in a focused direction.
Imagine somebody speaking in a direction in an open, flat field, and you standing a few meters behind them.
- when they speak, you will probably hear them somehow clearly;
- when speaking into the megaphone, keeping it close to their face, you will hear them much less than before; if there is a vertical obstacle large enough (like a building) a few meters in front of you, you'll hear the reflected sound much more clearly;
That's because when the megaphone is put right in front of their mouth, the sound is forced to a specific direction.
It would be almost the same as they were speaking with their mouth next to a hole in a wall: if you're standing behind them, you'll hear almost nothing, but if you're on the other side of that hole, you'll hear them very clearly.
Sound in a chapel or church
Again, not amplification.
Chapels and churches are large structures with big and very reflective surfaces (concrete walls/domes/arches, glass windows, marble floors, etc) that are both flat and curved (concave): a flat surface reflects sounds in the opposite direction, while a concave curve focuses it.
The sum of that combination results in focused sounds easily reflected by flat surfaces, and since some of those reflections arrive at very close intervals, you get a "louder" sound because of the sum of those vibrations arriving almost at the same time.
One common effect of similar structures is that known as "whispering-gallery waves": when one whispers in a very specific point of the structure, that whisper can be clearly heard at another very specific point which is sometimes quite far away, even if, in normal condition, that distance wouldn't allow the listener to hear the original sound as clearly.
The concept is similar to that of churches, but with less reverberation: the sound waves are better and uniformly propagated in the whole space, allowing audiences to clearly hear any sound coming from the stage even at 30-40 meters away. Specific architecture elements are used to provide better sound propagation, including special panels put on walls or hanging from the ceiling, shaped in ways that improve sound reflections.
Most of the sound of instruments having sound boards doesn't come directly from the strings, but from the vibrations of the sound board itself.
The vibration of the strings passes through elements that are able to resonate (vibrate at similar frequencies): the bridge, the sound/soul post, and the sound board itself.
The hollow space, type of materials and their thinness used in sound boards all contribute in a better efficiency of sound propagation, due to their matching the acoustic impedence. Simply put: they are able to get a better performance of the kinetic energy coming from the vibration of the strings, transforming most of it in air vibrations instead of dissipating that energy.
There are other ways of amplifying sound, not involving electricity. As it's now quite clear, we still need power coming from another source, though.
One example is the pneumatic amplifier.
It used a simple mechanism: a diaphragm put at the end of a "microphone" cone would transmit the vibration through a valve, put within a channel that had pressurized air coming from a pump: the increased air pressure, finally exiting through an output cone, made the sound much louder than its original source. It obviously wasn't very accurate, but the concept was quite ingenuous.
Here is a diagram of its mechanism.