A ♯ raises a note by a semitone or halftone.
I'm confused. If E and F are a halftone apart, why can't F be E♯?
It can be depending on the context . If you were using the F♯ major scale, you would have the notes
F♯, G♯, A♯, B, C♯, D♯, and E♯. Another common example is in a C♯ major chord you would have the notes
C♯, E♯, and G♯. The E♯ is an enharmonic equivalent to F. F is used a lot more though, since it is a naturally named note. In the same way, F♭ can used to describe E.
How you name a note being used is typically determined by the scale and context. For example : if you are going up the chromatic scale you would be using sharps to name the "black keys" and if you were going down the chromatic scale you use flats instead. There are a lot of things to think about when naming a note and you can see more examples of how notes are named in this question.
Sometimes indeed it HAS to be E#. It will depend on what key you're in as to what it is called. If a key has 6 sharps, the order is F#, C#, G#, D#, A# and E#.It's a technicality, but as far as writing it down is concerned, E# would be found on the 'E' place on the stave. It won't be found in the 'F' place, because that would signify an F note, which is already sharp. It could be naturalised, but that gets even more confusing, believe me ! There are other reasons why it needs to be called E#, all technical. As Dom says, the same sort of things happen to other notes as well. That's before we start on about 'is it G or F## (Fx)?' etc.
Maybe you had the idea that the black keys on piano were # and b, but that's only partly true. On guitar, there are no black keys, so maybe that's where the question came from.
A simple rule for major scales is that each letter name must be used once, and once only (until you repeat and start again !). Another rule is that the interval between successive notes of the scale must be
Tone - Tone - Semitone - Tone - Tone - Tone - Semitone.
Using this rule, the notes of a C major scale are C D E F G A B C
This is because there are additional tones between C and D, D and E, F and G, G and A and A and B. Look at a piano keyboard and you will see this - in the above example it just so happens that the notes of the C scale are all the White notes, and the interspersed intervals are the Black notes.
For more 'complex' scales we use the above 2 rules, and denote any black notes that we use as either 'Sharp' or Flat. Thus the black note between the F and the G can be denoted as either F Sharp (F#) or G Flat (Gb) depending on the context in which it is being used.
To get back to the original point (!) if we take the scale of F#
We must use all the letter names
therefore the notes of the scale become F#, G# A#, B, C#, D#, E# and F#
There has to be an E# because there must be an E something, and there can only be a half tone between that something and the F# - therefore the E (which is a whole tone below the F#) must be 'sharpened' - i.e. lifted by a half tone. There is no black note between E and F - therefore the white note next to the E is used (normally thought of as 'F') but in this case it’s E#.
It’s nothing more than spelling !!
A practical example to go with the valid theoretical answers: If one needs to play both F and F# on a harp, the harpist is going to have the harp set to E# and F#. Harps don't have any accidental strings, to get accidentals a pedal is set changing all (but the few lowest) strings to the accidental. So, if the F strings are made F#, the only way to get F is to make E# if both tones are needed at once.