Also known as bebop, Bop was a radical new music that developed gradually in the early '40s and exploded in 1945. The main difference between bop and swing is that the soloists engaged in chordal (rather than melodic) improvisation, often discarding the melody altogether after the first chorus and using the chords as the basis for the solo. Ensembles tended to be unisons, most jazz groups were under seven pieces, and the soloist was free to get as adventurous as possible as long as the overall improvisation fit into the chord structure. Since the virtuoso musicians were getting away from using the melodies as the basis for their solos (leading some listeners to ask "Where's the melody?") and some of the tempos were very fast, bop divorced itself from popular music and a dancing audience, uplifting jazz to an art music but cutting deeply into its potential commercial success. Ironically the once-radical bebop style has become the foundation for all of the innovations that followed and now can be almost thought of as establishment music. Among its key innovators were altoist Charlie Parker, trumpeter Dizzy Gillespie, pianist Bud Powell, drummer Max Roach, and pianist/composer Thelonious Monk.
It has become increasingly difficult to categorize modern jazz. A large segment of the music does not fit into any historical style; it is not as rock-oriented as fusion or as free as avant garde. Starting with the rise of Wynton Marsalis in 1979, a whole generation of younger players chose to play an updated variety of hard bop that was also influenced by the mid-'60s' Miles Davis Quintet and aspects of free jazz. Since this music (which often features complex chordal improvisation) has become the norm for jazz in the 1990s, the terms modern mainstream or Post-Bop are used for everything from Wallace Roney to John Scofield, and symbolize the eclectic scene as jazz enters its second century.
It sounds like a tough task to categorize music this way when the distinction is so fine and vague. You might have better luck broadening your categories if you're organizing your CDs this way.