Nietzsche names these two styles the Apolline and Dionysiac (or Apollonian / Dionysian) after two important, complex, and radically different Greek gods:
For Nietzsche's Birth of Tragedy, Apollo as the God of the Sun and prophecy symbolizes the Apolline style of light, order, & individuation of form as seen in dreams and epic poetry (e.g., Homer's Iliad and Odyssey). Apollo was also the god of medicine, all fine arts, esp. music (including the lyre), eloquence, archery, and care of flocks and herds.
Apollo represents a model of moral excellence with a beneficent and elevating influence. As a god of justice, Apollo discouraged vengeance and encouraged purification and penance as a means to expiate crime, as in Aeschylus's The Euminides. See also Shelley's Hymn of Apollo (1824). In Birth of Tragedy Apollo represents the illusory forms of dreams and art with which human audiences identify, investing these figures with enduring life and special meaning.
Dionysus as the God of Wine and festivals symbolizes ecstasy, intoxication, fertility, sexuality, masks, and the individual's absorption into a group or nature. He is the youngest of the Olympians and the only one with a mortal mother. For Birth of Tragedy Dionysus represents—in contrast to the elevated, individualized forms of Apolline dream-art—the formless immersion of the individual into the whole of society or nature through lyric poetry of loss, longing, or denunciation. The individual self willingly dies for unity with some larger identity, being, or force.
Birth of Tragedy puts the attributes of the Apolline and the Dionysiac in a dialogue or dialectic that is not merely oppositional. Instead, dialectic exchange and interaction make the two figures enhance and inform each other, so that each offers what the other cannot, and together they create a whole picture or system of the tragedy, art, or the world.
In other words, you don't choose between Apollo and Dionysus but recognize them as essential, complementary aspects of human personality or identity. Dionysus has "many close connections and shared aspects with Apollo," Marcel Detienne writes in the Encyclopedia of Religion (358). At Delphi, the center of Apolline prophecy, a niche was dedicated to Dionysus, who was himself a god who inspired prophecy at other sites.
For Nietzsche, tragedy represents the greatest art form because there both impulses meet and co-produce each other at once in the ultimate aesthetic experience of the sublime—or, in Aristotelian terms, catharsis.
For the Dionysiac impulse, Nietzsche sometimes substitutes Silenus, Bacchus, or the Satyr. In Greek mythology, these figures all share traits or appearances with Dionysus and sometimes are identical with him. (One way they're Dionysiac is by losing their individuality or sharing a common identity with the divine.)
Marcel Detienne, "Dionysos." trans. David M. Weeks. Encyclopedia of Religion, ed. Mircea Eliade. NY: Macmillan, 1987. 358-61
Sir Paul Harvey, ed. Oxford Companion to Classical Literature. 1940.
J.E. Zimmerman, Dictionary of Classical Mythology. 1964. NY: Bantam, 1971.
"Apollo vs Dionysus: The Only Theme Your Students will ever Need in Writing about Literature" by Michael Thro, VCCA Journal 10.2 (Summer 1996): 11-18