I am studying the normal modes and I am confused because there are several names that I am not sure if they are the same physical phenomena or they are different things that are related. I would like to know the definitions of each one and how these concepts are related to the resonances of a room. I read the chapter on standing waves in the 9th edition of Serway and Vuille's book and I know that standing waves are waves created by the interference of two waves in opposite directions and produce nodes and antinodes. This book explains this phenomenon with a string model, but this model is a bit different from a room model and doesn't mention normal modes, so how does this relate to normal modes, room acoustic resonances, room modes, and forced vibrations?
All mechanical systems (and you could consider an acoustical system as a mechanical one. This is justified in the use of electro-acoustico-mechanical analogies) have at least one preferred way of "behaving" when oscillating. This has to do with their normal response when they are excited too, which is of course the topic of forced excitation. These terms are clarified below.
Quoting Harris' Shock and Vibration Handbook:
Mode of vibration: In a system undergoing vibration, a mode of vibration is a characteristic pattern assumed by the system in which the motion of every particle is simple harmonic with the same frequency. Two or more modes may exist concurrently in a multiple-degree-of-freedom system.
This means that, according to the authors, a mode is the pattern of vibration. This is consistent with the fact that for a specific geometry, you will always (in the linear regime) get the same pattern for a specific mode, regardless of the frequency that excites the mode. For example, the modes of a rectangular plate or a parallelepiped room have the same patterns (different for each case of course) no matter what their scale is. Of course scale does affect the frequency that excites the mode. Based on that and quoting the same book again
Natural frequency: Natural frequency is the frequency of free vibration of a system. For a multiple-degree-of-freedom system, the natural frequencies are the frequencies of the normal modes of vibration.
and the normal mode of vibration is (again quoting the same reference)
Natural mode of vibration: The natural mode of vibration is a mode of vibration assumed by a system when vibrating freely.
Key word here is freely. This means (as mentioned in the beginning) that the natural modes (excited at the natural frequencies) are the ways (at the frequencies) that the system "likes to" (or has the "tendency") to vibrate. Since acoustics, like vibrations can be described by the "unified framework" of oscillations the same principles apply there too.
Now, according to the same authors, there is a specific definition for the normal modes and this is
Normal mode of vibration: A normal mode of vibration is a mode of vibration that is uncoupled from (i.e., can exist independently of) other modes of vibration of a system. When vibration of the system is defined as an eigenvalue problem, the normal modes are the eigenvectors and the normal mode frequencies are the eigenvalues. The term classical normal mode is sometimes applied to the normal modes of a vibrating system characterized by vibration of each element of the system at the same frequency and phase. In general, classical normal modes exist only in systems having no damping or having particular types of damping.
Now, coming to resonances. As I have already done multiple times in this answer I will, once more, quote the same old reference for a definition of resonance. This is
Resonance: Resonance of a system in forced vibration exists when any change, however small, in the frequency of excitation causes a decrease in the response of the system.
For the sake of completeness I will also quote a term that is not used so often. This is antiresonance and according to the same source it is
Antiresonance: For a system in forced oscillation, antiresonance exists at a point when any change, however small, in the frequency of excitation causes an increase in the response at this point.
As you can see, when we refer to resonances we are already talking about forced excitation of a system. As you may already know, when we are forcing oscillations to a system we can "arbitrarily" pick the frequency of excitation. The frequency at which a resonance exists/happens is called resonance frequency (or frequency of resonance). The same definition from the source is
Resonance frequency: Resonance frequency is a frequency at which resonance exists.
Now, coming to the part we didn't touch so far, the standing waves. According to the reference, a standing wave is
Standing wave: A standing wave is a periodic wave having a fixed distribution in space which is the result of interference of progressive waves of the same frequency and kind. Such waves are characterized by the existence of nodes or partial nodes and antinodes that are fixed in space.
Of course you may already know that, but on the other hand you should consider how would it be possible for a system of infinite extent to have natural modes (free vibrations) or exhibit resonances (forced vibrations). I don't think you could have such a thing (please correct me if I am wrong). This means that in a sense you can think of the standing waves as the natural/normal modes of an acoustical system. These two "things" (modes and standing waves) seem to have similar characteristics (according to the definitions presented above). Both exhibit fixed distributions in the spatial variables which constitutes them, at least for practical reasons, similar if not identical.
To explain the normal modes (answering to your comment), you first of all have to consider that they are modes. As stated above, a mode is a specific spatial distribution pattern (in the room acoustics case, most often of pressure or displacement but it is not restricted to only those quantities).
Now, to make it clearer, modes are excited at specific frequencies, no matter if we are talking about free or forced oscillations. The only difference being that when a system is freely vibrating it does so only at the normal frequencies (in the stade-state) whereas in forced oscillations it vibrates at the frequency of the external force. Nevertheless, when the frequency of external forcing function coincides with one of the natural frequencies you achieve resonance and you can observe the mode. If those modes (regardless of whether they are excited at a freely or forced vibrating system) are (linearly) independent then we talking about normal modes.
As already mentioned in the answer, the normal modes are the eigenvectors and the frequencies they happen are the eigenvalues of an oscillating system is stated/modeled as an eigenvalue problem.