The F-clef is named so because the line passing between the two dots corresponds to the note F. The symbol 𝄢 evolved from the Latin letter “F”, its two horizontal strokes reduced to dots.
To me, 𝄢 is a more elegant symbol than 𝄞 (G-clef). The dots aside, 𝄢 looks very much like a golden spiral, i.e. a spiral whose growth factor is the golden ratio 𝜑. This inspired me to make the F-clef from an old gold-plated guitar string:
When placed on the fourth line of the stave, 𝄢 is called the bass clef. The bass clef is the only F-clef in modern use.