In music theory, how many semitones make up an octave?

Twelve, in the standard Western system - and the number is a compromise rather than a discovery. The physics gives you the octave for free, since doubling a frequency sounds like the same note, but it does not tell you how to divide the space between.

The trouble is that stacking pure fifths never lands back on the start. Twelve of them overshoot seven octaves by a small amount, about a quarter of a semitone, known as the Pythagorean comma - so a keyboard tuned in perfect intervals is in tune in some keys and audibly foul in others. Equal temperament fixes this by making every semitone identically slightly wrong, dividing the octave into twelve equal steps, so that nothing is perfect and everything is usable in every key.

Other systems answer differently. Classical Indian and Arabic music use smaller intervals that a piano cannot play at all, and a good deal of what sounds exotic to a Western ear is simply notes that fall between its keys. The compromise is also why a string quartet or a choir, with no fixed pitches, tends to drift towards purer intervals than a piano can produce.