There are many various film and digital sensor sizes and formats.

135 film (also known as "full-frame") became the most used film in the 60s and the most common format was 3/2, meaning 36mm x 24mm frame size. The camera determined the format/ratio but was always 24mm high. There were even half frame cameras that used this film but would result in 18mm x 24mm frame sizes and would fit almost twice as many photos in one roll.

EDITED PHOTO FROM WIKI AND ABOVE LINK

APS was also a film size and the classic, 3/2 ratio/format, had a size of 24mm x 16mm (roughly).

For APS film, the height (red side) of the film was/is 16mm. This is where the "crop factor" comes in.

16mm x 1.5 = 24mm

And also relates to the field of view. If you have a 50mm lens on a full frame camera but placed it on an APS-C camera, you would get a different (cropped) field of view. So a 34mm lens, x 1.5 = 50mm (actually 52.5, but I'm rounding it down. REALLY rounding it down...).

In summary, if you had 2 cameras. 1 with ASP-C with a 35mm lens and a full frame with a 50mm lens, they would almost be seeing exactly the same thing.

Regarding surface area, yes, there is even more reduction.

So:

36mm x 24mm = 864mm² (full frame)

24mm x 16mm = 384mm² (APS-C)

As a percentage of area, the APS-C only covers 44% of the full frame. This is not the "crop factor".

EDIT: I'd like to add that digital cameras are capable of all sorts of different aspect ratios now which "crop" the sensor being used, anyway. It, in turn, changes the surface area yet will still be referred to as a FF or ASP crop camera.