Have you ever been in a musical environment, perhaps a class, lesson, rehearsal, or jam session, and heard someone use the term "enharmonic"? What does the term "enharmonic" mean?
"Enharmonic" is a fancy word that means something quite simple - an alternate musical spelling. We use the term "enharmonic" in music when we want to point out that there are two ways to indicate the same note, interval, or scale.
Let's take a look at an example of each.
Enharmonic spellings can be used to indicate different names for the same note.
Notice that enharmonic spellings can be used on white notes (of the piano) as well as black notes. Black notes can have a "sharp" name as well as a "flat" name. White notes can also have two names: their natural name and either a sharp or flat name.
Enharmonic spellings can be used to indicate different names for the same interval.
Because the F# and Gb are enharmonic spellings of the same tone (in the example above), the interval of the augmented 4th is audibly no different from that of a diminished 5th.
Because the D# and Eb are enharmonic spellings of the same tone (in the example above), the interval of the augmented 2nd is audibly no different from that of a minor 3rd.
Enharmonic spellings can be used to indicate different names for the same scale.
F# major is exactly the same scale as Gb major, the only difference being the alternate spellings of the degrees of each scale. In other words, the 1st, 2nd, and 3rd degree of the F# major scale are F#, G#, and A#, while the 1st, 2nd, and 3rd degree of the Gb major scale are Gb, Ab, and Bb. Generally speaking, F# major and Gb major are the only scales that are commonly notated using enharmonic spellings. Both have 6 sharps/flats in the key signature, so most musicians or composers will make the choice as to which scale to use based on personal preference. Other scales can be written using enharmonic spellings, but often the need for things like double-sharps or double-flats makes the enharmonic spelling too confusing and cluttered to be easily read.
Consider the C major scale and its absurd enharmonic spelling, the B# major scale, which you would never encounter in a real musical situation because of how overly complicated and confusing it is to read and notate.