By my understanding, "drama" here refers to the action that propels the plot forward and keeps the story moving in a logical and exciting sequence of events that ensures we stay on the edge of our seats (think action movie: all drama, big event after big event). By "narrative," I understand "exposition"—a character's narration of events that we don't actually witness; the character just describes them to us (think monologues).
Sophocles's Oedipus Rex is a story that strikes an unusual balance between drama and narrative. The style in Sophocles's day was to keep the time covered by a play relatively short—all the play's events should elapse in 24 hours or less, and take place in more or less the same geographical space. Oedipus's tale traverses many years and several cities; but Sophocles, in true Ancient-Greek fashion, shows us only a day of Oedipus's life. Through the narration of different characters we must listen to the events of the past rather than witness them. Examples of this occur when the messenger tells Oedipus how he was given a child by a shepherd, and when the shepherd arrives to recount how he saved the baby he was told to abandon.
Oedipus Rex is a play heavy in narration and exposition, but there are scenes where the action (or drama) progresses before us on the stage, propelled by the decisions characters make in the present. For example, at the play's conclusion, the blind Oedipus addresses his citizens and Creon leads him out of the city.