To the average Western listener, medieval European chant and classical Indian music are the two most familiar traditions that are not based on major and minor scales. But many other musical traditions around the world are not based on Western scales. Some of these have modes similar to the medieval Church modes; they also tend to be a list of notes (or a pattern of intervals) used with a specific finalis, which may encourage certain types of melodies. While the church mode/jazz mode tradition features diatonic modes (which can be played using only the white keys of a piano), non-Western modes may use other types of scales.
In other music traditions, modes are much more like Indian ragas, featuring important variations in tuning and melodic expectations from one mode to the next, so that each mode may be seen as a collection of related melodic ideas, phrases, and ornamentations that are traditionally played with a certain set of notes tuned in a certain way. (Some non-Indian traditions even use the term raga.)
All of these musics have long traditions that are very different from the familiar major-minor tonal system, and usually also have a different approach to harmony, rhythm, and performance practice.