Even though you are learning the flute, I think it is worth understanding the piano keyboard, and thinking about key signatures in terms of that. It's worth experimenting with a piano, or an electronic keyboard, or even web app like this one.
The white keys are the 'natural' notes A-G. The black keys are used to play sharps and flats. Notice that B flat is the same piano key as A sharp, and so on.
There is no B#/Cb or E#/Fb -- that's just one of those things; I suggest getting used to the basics before trying to understand why this is.
The first scale most people learn is C major. That uses just the white keys, with no sharps or flats.
You'll find you can play lots of simple Western melodies using just the white keys. Try Twinkle Twinkle Little Star - C,C,G,G,A,A,G F,F,E,E,D,D,C
That melody begins and ends on C; but more generally it 'feels' as if C is the 'home' note. Try ending on a different note -- somehow the tune feels unresolved. So the melody is 'centred' on C.
So this is Twinkle Twinkle in C major because:
- C feels like the home note
- It uses only the white keys, which is the C major's key signature
Now let's try playing Twinkle Twinkle in D. What happens if we just move all the notes up by one white key? D,D,A,A,B,B,A G,G,F,F,E,E,D
Actually it sounds great, until you hit the F. The F sounds wrong. You can tell why, by counting the black keys between notes. When we played that part of the melody in C, it stepped from F to E - that's one semitone because there is no black key between F and E. Now we're stepping from G to F there's a black key inbetween, making it two semitones.
To fix that, play an F# instead. D,D,A,A,B,B,A G,G,F#,F#,E,E,D
What we've just done is called transposing - moving a piece of music from one key to another. We've transposed Twinkle Twinkle from C major to D major.
Twinkle Twinkle doesn't contain the 7th note of the scale, but you can see that the step from A to B is two semitones, and the step from B to C is only one semitone. So to fix that, in D major we play C# instead.
So - to transpose something from C major to D major, move everything up by one note letter, play F# instead of F, and C# instead of C.
When writing music on a stave, it gets messy having to put sharp and flat signs every time a black note is played. Instead they are marked at the beginning of the piece, just after the clef. If we were writing down our version of Twinkle Twinkle in D major, we would write the clef, then a sharp sign each on the F and C lines, then the time signature.
It turns out that no two major keys have the same combination of sharps and flats, so a musician will look at those two sharp signs, and say "aha, D major" -- so that notation is called a key signature, because it is an easily recognisable indication of what key we are using.
But it's not quite that simple, because we also have minor keys. A minor has no sharps of flats, just like C major. The only way to differentiate is to recognise that the melody is centred on A. Likewise, B minor has the same sharps and flats as D major.
There are more scales too -- but that's a more advanced topic, to be tackled when you're comfortable with these basics.
All of this is exactly the same on your flute; it's just not as visually obvious as on a piano keyboard.