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As a matter of fact, our parish music director included the Ars Antiqua composers in his survey of Western Church Music (for which I received one college credit! with which I can do nothing!)

Basically this style of music grew out of chant. Singers started by singing the chant melody in octaves (pretty inevitable when men and boys sang together), then started singing the same melody in parallel but on a fourth or fifth interval (this is sometimes called "parallel organum". Then composers invented the idea of counterpoint by taking the original chant melody and singing it v-e-r-y s-l-o-w-l-y in one voice part (the cantus firmus, usually carried by the tenors), and singing variations (usually a completely different melody or intensely ornamented versions of the cantus firmus) on top of it.

Leading exponents of this style hung around Paris, particularly the Cathedral of Notre Dame and its choir school. Many of the compositions we have are anonymous, but the best known composers of the period are Leonin and Perotin (who always seem to be bracketed together like Abbott & Costello), Franco of Cologne, Pierre de la Croix, and a German abbess, St. Hildegard of Bingen.

Here are some examples. It sounds very weird to modern ears -- the Ars Nova that came immediately afterwards was one of the big watersheds in music history:

Leonin, O dulce lignum

Perotin, Organum 4 Vocum

Anonymous, Alleluia/Ave Maria

St. Hildegard of Bingen, "Qui Sunt Hic" from Ordo Virtutem ("The Play of Virtues", an early morality play)