(1)…Focusing Screen

You can see image that you are going to expose on film, and focus on subject through the focusing screen. When subject is in focus on the screen, it will be exposed on film in focus.

(2)…Mirror

Mirror is located at 45 degrees angle behind the view lens. This mirror transports the image from the view lens to the focusing screen. And when the image comes on the screen, it will be horizontally reversed. This is because it is to be reflected at the mirror to reach the focusing screen.

(3)…Film surface

Almost the same image that you see on the focusing screen will be exposed on the film surface. But because the image is through the taking lens (and not the view lens), parallax occurs fundamentally. Most TLRs introduced after 1950′s is equipped with parallax correction mechanism.

(4)…View Lens

The image on the focusing screen comes from the view lens, and this lens is needed when you focus subject.

(5)…Taking Lens

The image exposed on film comes from the taking lens. This lens is coupled with the view lens, so when you focus subject through the view lens, it will automatically focus on the subject.

TLR in America

Around 1840, Daguerreotype had started to be manufactured in New York. Then, as the large-size TLR camera appeared in Europe, it was exported to America little by little, and around 1900, manufacturing was started there, too.
Twin-Lens Graphics Special, released in 1901, looked like a TLR camera, but it was not exactly Reflex camera, for the focusing screen was put on the camera back. On Twin-Lens Graphics, location of the focusing screen was changed for the camera to be a TLR.
In addition, Twin-Lens Graphics was a forerunner to the Graflex SLR camera.

Cloud Camera..

Around 1880, a staff of the meteorological observatory near London came up with the idea to build the mirror inside the camera to take a picture of cloud. That was when the camera employed the mirror in it for the first time.
Well, it is certainly effective because you just see forward to catch cloud above in the focusing screen.
This camera, therefore, was called “Cloud Camera”.
On this camera, two lenses were coupled with gear in the middle which syncronized the lense’ movement with the other, and the image on the focsuing screen standed up thanks to the mirror (before that, the image was up side down). This have been thought the direct root of TLR camera now.
But, there were still problems. One of them was that the camera was big, and not practical for that reason. And also, it was not for close photography because of parallax.

via Structure or TLR « History and structure of SLR, TLR Camera.