In Western music, 12-Tone Equal Temperament, or 12-TET, is the standard for defining where all of the notes lie. The octave is divided into 12 notes, where each note is 100 cents away from the previous and next notes. Quarter tones belong to the 24-TET system, or 24-Tone Equal Temperament. The octave is divided into 24 notes, where each note is 50 cents away from the previous and next notes. These are the quarter tones that appear between the chromatic tones that exist in 12-TET.
The quarter tones are notated using sesquiflat, semiflat, semisharp, and sesquisharp notation, as shown above. This style of notation was created by Myles Skinner in his quarter tone doctoral dissertation, where he analyzed works by Easley Blackwood, Alois Hába, Charles Ives, and Ivan Wyschnegradsky. Skinner’s style of notation is beginning to emerge as the standard for quarter tone notation.
If you are new to quarter tonality, Ivan Wyschnegradsky's 24 Preludes for Quarter Tone Systems is a great place to start.