An understanding of modal jazz requires prior knowledge of musical modes. Modes are the seven scales used in medieval music, but 'rediscovered' by composers like Claude Debussy and frequently used by 20th century composers. In be-bop as well as in hard bop, musicians used chords to provide the background for their soloss. A song would start out with a theme, which would introduce the chords used for the solos. These chords would be repeated throughout the whole song, while the soloists would play their parts. In the late 1950s, musicians frustrated with the ever repeated chords, tried the modal approach. They didn't write their songs using chords, but instead they used the modal scales. This meant that the bassist, for instance, didn't have to 'walk' from one important note of a chord to that of another - as long as he'd stay in the scale being used and accentuated the right notes within the scale, he could virtually go everywhere. The pianist, to name another example, wouldn't have to play the same chords or variations of the chords, but could do anything, as long as he'd stay within the scale being used. The soloist also had more freedom.
For example, one could ask the following question: let's say we'd play a song, and only use the Ionian scale in C (which contains all the white keys of the piano, in other words: no flats or sharps), aren't we just playing a non-modal song in C major, with the only difference that our song only contains one chord (C major)?
No, we're not. To understand this, one will have to know the difference between consonance and dissonance, which I am not going to explain here.
What's important to know is that it's possible for the bassist and the pianist to move to notes within the scale that are dissonant with the prime chord of that scale. For example: within the ionian scale explained above, the C is the prime note. Other notes, for example the B, are dissonant with C, so that they won't be used in a non-modal jazz song written in C, when playing the chord C. In a modal song they will be, which means the notes played will not be recognized as a part of C major.
In modal jazz, among the significant compositions were "So What" by Miles Davis and "Impressions" by John Coltrane. Both of these songs were in Dorian mode meaning they were in a minor mode.
They follow the same AABA song form and were in D dorian for the A sections and modulated a half step up to Eb Dorian for the B section.
In the album Kind of Blue by Miles Davis, one could see the exploration of modal jazz. It is a landmark album and is a jazz classic.
In improvising within a modal context, one would basically start by thinking that one must play the notes within that specific scale (e.g. D dorian- D,E,F,G,A,B,C,D) but one must remember that it is also possible to take several notes from that scale (and not all) to create smaller scales or note choices for improvisation.
For example, in D dorian, one may play the notes of the D minor triad ( In fact this is what Miles did in his improv to begin with) or one may even choose any of the triads available in that mode. (C maj, Dmin, Emin etc.)
The only thing is, if you choose a triad that is an upper structure triad of the chord you will get more tension in the line.
One may also use the many different pentatonic scales within the scale such as C major pentatonic, F major pentatonic and G major pentatonic. (Note that these are enharmonically A minor, D minor and E minor pentatonic respectively).
Such choices (that seem to limit the notes may further expand the improvisational choices for the player in approaching modal jazz).