In Manhattan, a duplex is a two-story dwelling within an apartment building—preferably pre-WWII, and even more desirable if designed by Rosario Candela or Emery Roth. It has a cachet just slightly below that of a "maisonette."
In the Middle America west of the Hudson and east of the Golden Gate, a duplex is a house split down the middle—or "divided against itself," in the words of Abraham Lincoln. It is presumably home to two honest working class families. The English euphemistically call it a "semi-detached house."
The Los Angeles duplex falls somewhere between the two extremes. It is neither for the filthy rich nor for the boorishly unglamorous. Instead, it's a charming, preferably Spanish-revival apartment building in which one family (however "family" might be defined on the West Coast) lives above the other.
Here are a few examples from current Craigslist posts.
915 S. Shenandoah.
"Heart of BH. Not BH adjacent."
Celebrated Los Angeles interior design maven Barbara Barry once lived in a duplex on 6th Street in the Miracle Mile district, the locus classicus of Los Angeles duplexity. And Shelley Winters, even after achieving success as an actress, famously lived with her mother in a Beverly Hills duplex.