The simpler the expression of two frequencies as a fraction, the more consonance, and vice versa for dissonance. Consonant notes blend together well; dissonance, not so much.

For example, A at 440hz, and an A an octave higher at 880hz. 440/880 = 1/2. An octave is extremely consonant. What's better? 440hz and 440hz played together = 1. A perfect unison is the most consonant sound ever. How about a fifth, A440hz to E660hz~? 440/660 = 2/3. Very simple, and explains why power chords are so pleasant.

Oh, but what about E330hz~ and F350hz~? 330/350 = 33/35... that's not a very elegant fraction, is it? That's what a minor second roughly is, and it's very dissonant.

That's roughly it.
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In applicable terms...

Perfect unisons/octaves + Perfect Fifths = Perfect consonances

Major and minor 3rds/6ths = Imperfect consonances

All 4ths, 2nds, 7ths and diminished/augmented intevrals = Dissonances

Dissonances want to resolve. Play a C major chord and play the major 7th along with it. You can hear the dissonance wants to resolve upwards to C.

Play a A sus4, the 4th sounds resolved when it moves down by step to C# (a consonance )