Why is singing/playing an instrument off-key a thing? Why are some frequencies of the acoustic wave okay and some not okay?
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I see two interpretations of your question, without understanding your knowledge level. Out of key could be referring to notes that are not in the scale or key the music is in. The alternative would be e.g. If the singer is attempting to sing the correct note but misses.
So the word you may be looking for is diatonic, which would be a note that is in the current key.
Music that is only consonant tends to be pretty boring. For example, a lot of children’s music uses only consonant, diatonic notes. There is nothing wrong with this, per se, but a lot of the things that make music interesting are when your expectations are broken. The song Mary Had a Little Lamb is a very good example of a song that has no dissonance in it. It goes up and down the major scale in a pleasant and completely unsurprising way.
Now, what is consonance and dissonance? It is a physical relationship between the tones in music. Tones that are simple ratios of each other sound consonant. That is, for example, a 400hz tone with a 300hz tone. That’s a 4:3 ratio. For every four cycles of one sound, there are 3 of the other. These will sound cinsonant together because they line up frequently.
By comparison, a 141:100 ratio is not a simple ratio, nor is it close to a simple one. So these will sound dissonant together.
If you’re singing and you miss a note by a tiny bit, our ears cover a bit and will agree with it and think it sounds fine, if not perfect. But if you sing too far off, rather than a nice simple ratio, you sing an ugly ratio, and it sticks out like a sore thumb. It’s even possible to miss a dissonant note and sing a consonant one by accident. If someone does this when singing a song you know, it’ll sound wrong even though it is consonant.
Almost all music has some dissonance in it. Some dissonance is gentle. Some is grating. Some is invigorating!* Some won’t even feel dissonant, but it might be an unexpected note or chord that adds some color or feeling to the music. Most is in between. You can’t just mash random keys on a keyboard together, because you’ll hit a lot more dissonant notes than sounds good together.
But dissonance used wisely gives you a feeling of tension. Tension is good. And then, usually, but not always, the song may resolve that tension by shifting to notes that are more consonant, and this will release that tension and feel good. Not all music is structured like this, of course, but most is.
For an example of some extremely dissonant music made to sound quite beautiful, here is Eric Whitacre, who writes choral music that uses a lot of chords that have a bunch of “wrong” notes – but he does this to great effect in this cover of a familiar song, which itself has quite a bit of dissonance in it too:
For an example of some invigorating or whimsical dissonance, listen to the Simpsons theme song. When. They sing “the SIMPsons”, that “simp” is a dissonant note that would be out of key in the major scale. If they sang the “right” note, it would sound a lot more boring. The bass line also uses that same dissonant interval too.
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