Narrative essays will always tell a story. Descriptive essays are focused on clearly expressing the characteristics, qualities and sometimes the appearance of a person, place, idea, political/social movement, etc.
The key idea for writing a narrative essay is that of the narrative.
"The term is usually applied to anecdotes, exemplums, fables, fabliaux, fairy tales, incidents, legends, novels, novelettes, short stories, and tales" (eNotes).
In a narrative essay, a chronology or a series of events must be present. For example, a narrative essay about a basketball game might relate the story how one team came from behind to win the game by a single point, hitting a buzzer beater at the final second.
A descriptive essay, in contrast, might discuss a basketball game without mentioning anything about what happened in the the game and instead focusing on the sights and sounds and feelings of the game. A descriptive essay on this topic might describe the sound of the buzzer at the end of the game or the size of the arena where the game takes place.
The crucial difference in these two essays on a basketball game is that the narrative essay tells the story of the game and the descriptive essay attempts to express the details of the game regardless of action or outcome, showing instead the characteristics of an experience of the game.
To offer an analogy, a descriptive essay can be like a snapshot photograph, capturing a single moment in detail, while a narrative essay will be more like a movie, relating a series of events.
Narrative essays and descriptive essays are not necessarily at odds with one another. These two forms of the essay can overlap considerably. A narrative essay can feature a prominent use of detailed descriptions to bring its story alive. And a descriptive essay focused on offering a full rendering of a person, place or thing might use stories to help complete that rendering.