In a nutshell, the term enharmonic equivalent means notes that sound the same as one another but are named or “spelled” differently (and this concept can also be extended to include intervals and scales).
The musical alphabet consists of seven main notes represented by seven letters: A-B-C-D-E-F-G. However, you might be aware that there are twelve possible notes (half steps) in an octave. For example on the guitar, you reach “E” again on the low (or high) E string once you get to the 12th fret. On the piano, you can count 12 keys if you play every key from A to A, C to C, or any other octave for that matter. Adjacent frets are half steps, and so are adjacent keys on a piano.
In order to name all of these notes, we have sharps (#) and flats (b) in music notation. To “sharpen” a note means to raise it one half step—so A# is one half step above A. It is the note between A and B. Similarly, to “flatten” a note means to lower it by one half step—so Bb is one half step below B.
Here is where we reach the topic of enharmonic equivalents. In the example above, I moved up a half step from A to reach A#, and I also moved back a half step from B to reach Bb. These two notes sound identical. They are the same; one key represents these two notes on a keyboard, and so does one fret on a guitar.
These notes are called enharmonic equivalents because they sound the same—indeed they are the same note—they just go by different names depending on the situation. G# is the same as Ab, C# is the same as Db, F# is the same as Gb, and so on. *Note: B to C, E to F are separated by half steps, so B#=C, and Cb=B, etc.
The reason we use two different names for these seemingly identical notes comes from the way we conventionally discuss music theory. Traditionally, if one is composing music in some key, it is only acceptable to have one “kind” of each note name. Sounds confusing, but here’s an example. In the key of G major, our notes are G-A-B-C-D-E-F#-G. We name the seventh note F# because if we named it Gb, we would have two different types of G (G and Gb), and that is a no-no. We want only one kind of each note (by kind I mean natural, sharp or flat) to keep things organized and logical. As is always the case in music, there are exceptions, but this convention should be observed whenever possible.
That being said, it is sometimes easier to think of a note as being the sharpened version of some note as opposed to the flattened version of another (and vice versa), even if it is not notated as such.
Generally, if we are moving up a scale or melody, it is easier to think of sharpening notes because you simply move up from a note you are already on. Similarly, if we are moving in a downward direction it is often easier to think of some note as being flattened. You will still arrive on the correct note, but it sometimes is easier for our brains to process them this way.
That’s all for now—hopefully this helps to make sense of a somewhat tricky concept. Rock on!