Probably the crucial concept you need to know about here is that of flange focal distance - for any given lens system, commonly called a "mount" (examples would be the Nikon F mount and the Canon EF mount), the lens must be positioned a precise distance from the sensor (46.50 mm for the Nikon F mount and 44.00 mm for the Canon EF mount). The tolerance on this distance is really very small - typically of the order of hundredths of a millimeter (or tens of microns if you think in those terms); if your lens is out by more than that, you won't get sharp images.
In practical terms, what this means is that you can't fit an arbitrary lens to an arbitrary camera unless you happen to have access to much higher precision engineering kit than most people have, even if you could get the current lens off your Finepix camera / phone / webcam / whatever.
On the other hand, fitting a lens designed for one interchangeable lens mount (e.g. Nikon F) to a camera with a different interchangeable lens mount (e.g. Canon EF) is possible. In the cases where you want to fit a lens with a longer flange focal distance (e.g. Nikon F at 46.50 mm) to a camera with a shorter flange focal distance (e.g. Canon EF at 44.00 mm), various companies will make you a precisely machined adapter which just mounts the lens the precise distance further away from the sensor (2.50 mm in the case above) than it would for a "native" lens.
The more difficult case is mounting a lens with a shorter flange focal distance to a camera with a longer flange focal distance as you can't get an adapter to mount the lens inside the camera body! This kind of adapter needs to have extra glass in it in order to cause the lens to focus at the right point, which generally makes them much more expensive than the simple adapters referred to in the previous paragraph.
Also, in both cases, you will tend to lose any electronic control over the lens when you're using an adapter, unless it's an official adapter designed for converting one lens mount from one company to another lens mount from that company (e.g. Nikon produce an adapter for mounting a Nikon F lens onto a Nikon 1 mount camera).
One other thing you'll need to consider is that of imaging circle: simplified very slightly, a lens is designed to work with a specific size of sensor. Using a lens designed for a smaller sensor on a camera with a larger sensor will result in the image circle not covering the edges of the larger sensor.