What physical process happens when two sounds occur at the same time?
When two or more sounds happen at the same time, their sound waves (commonly depicted as a graph of increasing and decreasing air pressure) superimpose into one composite waveform, through "wave addition". For each tiny moment in time, the air pressure is the sum of what it would have been from each instrument alone.
Wave addition is how the speaker in your headphone can play bass and guitar at the same time (and drums, voice, etc). If the waves being added are identical, the signal doesn't change but just gets louder. Even more unusual, when the signals add up to zero they may cause silence. This is extremely unusual with real instruments, but it is what causes "beats" when two instruments play the same note just slightly out of tune with each other.
Why do some combinations become music but not others?
Well, if the combination has the ability to communicate to the listener a musical idea, then it's music. If there is too much noise for the listener to distinguish what's being played, or if there aren't really any ideas being sent, you won't hear music.
Some common musical ideas are melody, harmony, and rhythm, as well as "color", imitation, and mood.
What about harmonics and overtones?
When two frequencies sound good together we say they are "consonant", as opposed to "dissonant". Real instruments don't produce a single frequency sound but an assortment of frequencies ("overtones") which are all added together physically, and which are more or less related mathematically to the pitch (or fundamental frequency) the instrument is playing.
Math and physics have identified a sequence of frequencies called "harmonics" (or "modes of vibration") which generally sound good together. These are known to all brass (i.e. trumpet, trombone) players as the sequence of notes played at the same fingering going from low to high.
When an instrument or noisemaker has a composite sound that is mostly composed of the early harmonics in the sequence, we say that it has a "sweet" sound (or sweet timbre), like a bell. It's easy to communicate musical ideas such as melody and harmony using sweet sounds.
When superimposed tones (such as an out-of-tune ensemble or modes of an object's vibration) contain strong frequencies that are dissonant, we may call the sound "sour". Sourness obscures melody and harmony by making the intended fundamental frequency ambiguous to the listener and poisoning consonance.
Chords are primarily composed of notes in the harmonic sequence from some fundamental frequency, and many complex chords still do not stray from this rule, though they may contain higher harmonics. If a chord doesn't relate well enough to some fundamental frequency (or "tonic") via the harmonic series, it will sound sour or dissonant; but this is not necessarily bad: complex music frequently uses dissonance in producing tension or other desired qualities.
With classical instruments, the fundamental frequency is a lot louder than the other frequencies. The next loudest frequencies are nearby harmonics, which sound consonant to the fundamental frequency, with the higher harmonics more or less quieter. You know that B and C are dissonant, but because the volume of each next overtone is less than the previous, a classical instrument playing C is not dissonant with one playing E, even though B is audible as a harmonic of E... the B is less loud.
Percussive instruments like cymbals produce so many different frequencies that a person doesn't perceive a fundamental frequency. These instruments can communicate any musical idea that doesn't depend on tone (i.e. melody or harmony).
Further reading on overtones
Depending on what letter a person is pronouncing, a person's voice may be sweet or sour. For instance, the short vowels A and O (as in, "awesome" and "oafish") are very sweet. But the consonants by themselves (B, C, D, F etc.) have no fundamental frequency, and would make a sour sound if they were sustained. This is why holding the R sound is never advised for singers.
Interestingly, U as in umbrella (called /ɜː/ by linguists) has a strong overtone one octave higher than the strong overtone of E as in free (/iː/).
Further reading on voice
What is music?
No doubt this has been addressed elsewhere. But I would formulate it this way: Music is when a person uses sound to communicate ideas that are other than language.
Order and symmetry
Order and symmetry function like carrier waves so that listeners will perceive the ideas the musician is trying to communicate.
Pure symmetry and perfect order are boring but instantly recognizable in sound. The musician initiates the listener's recognition by order and symmetry; then whatever the musician wants to communicate is placed on top of the order and symmetry to be noticed instantly.