Hi Inge. It is interesting to see the various responses below. Mostly because the term Triplex means different things in various parts of the country.
Ms. Levy's answer is correct and exact. In New York City, a Triplex is an apartment in either a Coop, Condo or a Townhouse building consisting of three levels (or floors). Some of these apartments are the result of combining apartments. Others have indeed been built in this fashion.
These types of apartments can be quite expensive, especially in Manhattan. As a 2BR/2Bath Condo Triplex, I can suspect a price tag of well over a million dollars which adds a new wrinkle. The wrinkle is that in Manhattan we have a little thing called a "mansion tax." This is a tax imposed by NYC when an apartment sale exceeds $1M. The tax is equal to 1% of the purchase price. Of course, this poses no problem for those with a large amount of money. Good luck!