There are 12 notes in most Western music. When you double a frequency you go up an octave, but keep the same note.

Music is played in different “keys” though with 7-note scales, with letters assigned A-G. If you play the notes in order starting and ending at the letter for which the scale is named, then do the same for a different scale, the relationship between the notes will sound the same between the 2 scales, but your starting and ending pitch will be different.

Piano keys are arranged with all 12 notes being available, but arranged in the key of C-major or A-minor, where all notes are natural notes (no sharps or flats).

If you play just the white keys starting from C, you’ll be playing a C-major scale : C, D, E, F, G, A, and B. However, to play the F-major scale, you’re going to need to skip one white key and hit a black key: (F, G, A, Bb, C, D, and E). No letter repeats on a scale.

A sharp (#) or flat (b) note is just moving the smallest step you can to the right or left, respectively. For most notes, that’s moving to a black key, but there’s no black key between B and C or E and F sometimes it’s moving to another white key.

Why don’t we just ignore weird notes like Cb? Because every letter needs to be represented on a scale. Ab-minor, for instance has Ab, Bb, Cb, Db, Eb, Fb, and Gb. So even though Cb is the same frequency as B-natural, it serves the same role in the scale as E does in the key of C, and if you didn’t represent it as a flat note, your scale would have 2 "B"s and no “C.”

This gets even more important when you get into different instruments with different natural keys. A Piano, flute, bassoon, and other instruments are what we call “Concert C” instruments, which means they have the same natural key of C. However, other instruments are different.

A standard clarinet is a Bb instrument, meaning its Bb scale matches the C scale of a piano. You also have Eb-clarinets that are a little smaller, meaning that if they play a “C” they’ll be playing a concert Eb, which uses the same fingerings as a Concert Bb from a standard clarinet.

So when an Orchestra is playing something in the key of A-major (A, B, C#, D, E, F#, G#), an Eb-clarinet is playing in F#-major (F#, G#, A#, B, C#, D#, E#).

First, you don’t have to, it’s a useful convention.
Since the middle ages, the west has used a seven note scale with five whole steps and two half steps. This gives one scale, c major, with seven natural (neither flat nor sharp) notes.

As an aside, I believe there are six note scales.