Is there a specific word for a two-floored house?

When talking about family dwellings, they tend to be either on one floor (either a bungalow or a flat) or split across two floors. If a dwelling has two floors then it tends to be a house and not a flat but if, however, it is a flat split across two floors, it is called a maisonette.
Is there a specific word for a house (not part of a block of flats) that has two floors?
Is there a matching adjective?

I used to live in a bungalow but now we've moved into a [two-floored house]

This is generally called a "two-story house" in AmE. Someone might use a more colorful term (e.g. "New American") but I would view it as pure marketing propaganda, as I know of nothing in widespread use.

Do a search for "two-story house" or "2-story house" and you'll get all the results you can imagine.

The only related term I know is "duplex" which is really just a house -- often a two-story house -- that's been split into two separate units. These can be above each other or side-by-side, depending on the design.

(Edit) I'm reminded the British tend to spell it "two-storey" house, to distinguish it from the other kind of "story".

The original Anglo-Indian word bungalow, inside the UK is always a one-story house. The OED, with entries from the 17th century says,

Orig., a one-storied house (or temporary building, e.g. a summer-house), lightly built, usually with a thatched roof. In modern use, any one-storied house.

Outside of the UK, bungalow is often applied to a single or dual story house, where the second story is beneath the eaves, as @1006a has commented.

However for UK purposes, single story, is bungalow, any detached domestic residence that is more than that is called simply a house. Often people will say things like do they live in a house or bungalow?, expressing a distinction in the terms.

However, just to avoid confusion, the word house also tends to be used as a generic term for any domestic residence, where the nature of the structure is not known to the speaker - e.g. Do you watch much TV at your house? It could be a house, bungalow or flat in a sentence like that. It would be unlikely anyone would say Do you watch much TV at your bungalow? -even if it was known that they lived in a bungalow.

A flat in the UK is a residence with a shared front door, and common areas with other similar residences. It can be on any number of floors.

A semi-detached house is one that shares a party wall with a neighbour - so it is two houses in one structure, each with its own door. A terraced house is one that is in a row of three or more joined houses in a terrace.

A maisonette is chiefly a British term (also Australian) and expressed by the OED as A part of a residential building which is occupied separately, usually on more than one floor and having its own outside entrance.

Depending on the size of the house, BrEng has the following phrase:

two-up two-down

From the Cambridge Dictionary:

a small house on two levels that has just two main rooms on the ground floor and two bedrooms on the top floor.

The word 'duplex' is used quite a lot in the Indian Real Estate Market to describe a two storeyed house and even a two storeyed (split level) apartment.

In America, on the East coast, at least, there are several house styles that imply two stories: Gambrel, Colonial, Garrison, Cape Ann, Townhouse. All of these imply a reasonably full (as far as height goes) second floor. There are additional types which may have a second floor area that is constrained by roof pitch or other factors: Cape, Cape Cod. Generally speaking, a Split Level, Raised Ranch (or Split Entry/Split Foyer) is considered one story, as a flat Ranch house built to the same dimensions would have identical areas (only difference is each section's height above grade). A Bungalow in the New England area might have a very small and very height constrained second floor. The term Cottage in America could be anything from a beach shack to a Newport mansion from the Gilded Age. But, to be honest, house styles are rather in the eye of the beholder and local jargon and there are no hard and fast standards as to what constitutes them.