It's to do with the "circle of fifths". As you start from C and go up a fifth at a time you get the sharps:
C G D A E B F# C# G# D# A# E# B# F## etc
where of course normally B# is thought of as the same note as C and so the "circle" closes up. But in this context it is thought of as a different note.
As you go down a fifth at a time you get the flats
C F Bb Eb Ab Db Gb Cb Fb Bbb Ebb Abb Dbb Gbb Cbb etc ...
Again normally Dbb is thought of as the same note as C.
If you use pure harmonic fifths such as you get in the harmonic series, then each fifth (at 702 cents) will be a couple of cents sharper than the equal temperament fifth, and so the B# at the end of that first circle is noticeably sharper. There cents refers to a hundredth of a semitone.
Tuning by pure fifths is called the pythagorean tuning, so the difference between the B# and the C in this tuning is known as a "Pythagorean comma".
So in the Pythagorean tuning by pure harmonic series fifths, the tuning used for most music in the middle ages, then the B# and C notes actually differ in pitch. In fact as you continue further, next time around the circle, the A### when you get to it eventually also differs from C, and you can show quite easily that the circle never closes up exactly.
Our twelve tone scale is what is known as a "moment of symmetry" - if you start with a fifth, and keep adding more notes following the circle of fifths, then most times you have three different interval sizes. But every so often when it almost closes up you get a more symmetrical scale with only two interval sizes.
This happens with the pentatonic scale for instance, there are two interval sizes, a pythagorean whole tone and a pythagorean minor third. The pythagorean diatonic scale again has two interval sizes, the whole tone and the semitone.
We normally think of the twelve tone scale as having just one interval sizes the semitone. But in the pure harmonic series based pythagorean tuning, then it has two sizes of semitone - the diatonic semitone such as the one between E and F, and the chromatic semitone between C and C# say.
So anyway in a nutshell that's the theory behind it, you can go into it in a lot more detail.
So anyway if you start from a white note and go down by a fifth to a black note (or up by a fourth) and in the context wish to think of that as a pure fifth in the music, then the black note is functioning as a flat. If you go up to a black key by a pure fifth (or down by a fourth) then it is functioning as a sharp.
But the pitch is the same if you are playing in equal temperament. You hear a difference if you use a tuning with pure fifths or some other size of fifths.
Other sizes of fifths are also used. For instance if you want your major thirds to be pure harmonic series major thirds, then you need a fifth which is flatter than pure, because the pure major third is 14 cents flatter than the equal temperament one. So in the sequence C G D A E each fifth has to be a bit flatter than pure. The next tuning used historically after Pythagorean was quarter comma meantone which used fifths flatter than pure in order to achieve pure major thirds.
This is also tricky since you can't achieve an octave with three pure major thirds as in e.g. C E G# C - so something has to give. Usually the C-E would be tuned pure in quarter comma meantone and the Ab-C leaving the E - Ab as a wide major third.
Notice also here there's another distinction between sharps and flats, this time using pure major thirds instead of fifths. If you get to an Ab from C by going down by a major third then it is natural to call it a flat. But if you get to it from the E, then it would make more sense to call it a G#.
This is a different note from the Ab in the circle of fifths BTW, it is tuned differently since you get to it using major thirds instead of pure fifths. In short there are many tunings of sharps and flats in microtonal systems depending on the context and tunings used.
So - if it was just about fifths the subject would be relatively simple - but normally you are interested in other intervals as well, and may for instance want to trade off a somewhat flat fifth in order to achieve a purer (more mellow) major third.
Another disadvantage of quarter comma meantone is that with those very flat fifths, then the B# will be a lot flatter than the C (rather than sharper as it is in Pythagorean) - and in fact it is so flat that in medieval times the interval from the B# to the G was considered unplayable. So they dealt with that by moving that awkward fifth to a different place on the keyboard, so it was only ever needed in a key which they never played.
So quarter comma meantone is an example of a tuning with a rather large range of "key colour". Later on e.g. in Bach's time and later they explored many other tunings the so called "well temperings" which give each key a different tuning colouring - some of the fifths relatively pure and some not, ditto with the other intervals.
Many systems have been explored over the years. They are all basically compromises, because something has to give. Mathematically, frequencies that are a fifth apart are in a frequency ratio of 3/2, and if an octave apart obviously 2/1, and if a pure major third apart, 5/4. When you add musical intervals, you multiply the frequency ratios because we hear "logarithmically". Then the thing is that there is no way you can multiply any number of 5/4s together to get a power of 3/2, or any number of either to get a power of 2/1. So no interval made up of pure major thirds in any combination can be the same as an interval made up of pure fifths or an interval made up of octaves, so none of the "circles" of fifths, major thirds etc can close up if you use pure type intervals such as you find in the harmonic series..
The equal temperament is a possible solution if you want to be able to modulate to any key, have all the keys sounding identically tuned, and don't care too much about mellow major thirds and very pure fifths. We have come to find it very acceptable but in some historical times times the major thirds we accept as normal would have been considered rather "strident". In the middle ages the major third was regarded as a dissonance so they didn't try to tune it pure at all.