As I understand it, in fairly simple terms:
The quoted lens 'lengths' are usually 'equivalents' to the focal length of a single lens 'element'.
Imagine a Sherlock-Holmes detectives magnifying glass. That's a 'single element'.
Focal length would be how far you have to hold it away from your eye to get an in-focus image.
More magnification the lens offers, the further from your eye it 'focuses', longer the focal length.
Scientifically, focal length would be the distance from the viewing plane, when image in focus, and the nominal axis of the lens element. (given that lens is a bowed bit of glass of different thickness at the edges to the middle)
But few modern camera lenses are single element lenses; they have 'stack' of elements, some of which are likely to move forwards and back in the lens body to offer 'variable' focal length magnification, or 'zoom' as well as focusing, hence the 'Focal Length' as stamped on the side, is worked backwards.. NOT from the angle of view, but from the degree of subject to image magnification;
So an 18-55mm zoom lens is a lens that has the effective magnification of a single element lens of 18mm focal length at the wide end, and has the effective magnification of a 55mm lens at the telephoto end. And its incredibly unlikely that at any point in its operation, anything measures the actual mm the zoom ring is saying! ('Front' element on my 18-55 is aprox 90mm from the lens mount at the 18mm setting....95mm from the mount at the 55mm setting, and is shortest at the 35mm setting, when its aprox 85mm from the mount)
Angle of view? Well, most cameras take oblong pictures.... in landscape orientation, obviously the lens is giving a wider angle of view left to right, across the frame, than it is top to bottom. Focal length is fixed.. the angle of view set by the edges of the frame, in turn set by the size and proportions of the sensor/film behind it.
50mm lens on full frame/35mm is a 'standard' angle, i think about 100 Degrees or so inclusive. On a smaller frame APS/Crop size sensor, 50mm is a mild telephoto. On a larger sensor, medium format, its a wide angle....
As long as you don't encounter 'Vignetting'...
50mm lens for an APS/Crop sensor camera, likely to not have the 'spare' field of view that it would actually project an image over the entire area of a larger sensor; then, the barrel of the lens is likely to create a shadow at some distance beyond the dimensions of the sensor it was designed to project on, and the image would fade into that shadow where light is basically just masked.
So, a 50mm lens is always a 50mm lens... and always offers the same amount of magnification between scene and camera sensor... how much 'zoom' or what angle of view it provides, depends on the size of sensor its mounted in front of.
Which is where the Crop-Factor confusion arises, as long lenses DONT technically make the scene 'bigger'... they just don't make it as small!
The 'Taking' lens on a camera, shrinks a scene.. I don't know a 50ft high house, gets shrunk down to an image on sensor, that's perhaps only 25mm tall... if you shot film, you'd get a negative 24x36mm, and then 'enlarge' that image to print size, whatever that may be to view, perhaps 100x150mm, by shining light through the negative and ANOTHER lens, to project image onto the print paper. THAT lens in an 'enlarger' IS actually magnifying the image from the negative to the size you see. But still probably smaller than the scene to begin with.
Digitally that bit of the process is done with digits; the captured 'scene' reduced from whatever real world scale to be caught on a sensor, anything from 6x4mm up to around 24x36 in common cameras, as a grid of squares, assigned a brightness and colour value, that's then transcribed to whatever size screen you look at it on, divided into the same number of squares, and lit with recorded brightness and colour values. But still the same process; the recorded image is being 'enlarged' to the viewing size.
So to some degree, the focal length is actually a pretty 'meaningless' dimension... as on most lenses its NOT actually a real tangible measurable dimension, while its influence over what you are really bothered about... how big your subject will be in the frame of whatever viewing medium you eventually look at it in, is also a bit spurious, as that depends on how big the sensor behind the lens is, and what you do after capture, enlarging or cropping what you caught on the sensor.
And worth noting, as a reference 'associated' with magnification, its commonly only used in reference to cameras, where the lens is (more likely to be) only a 'part' of the over-all system, and not a complete user-ready 'product'.
Its a little bit of 'legacy', where the focal length was one of the more 'critical' features of the lens, from the days of the 'plate' camera, when it was literally little more than a light tight box, with a ground glass window on the back, and a hole for a lens in the front... you wanted a lens that focused on the distance from where you mounted it on the front, to where the ground glass view screen or exposure plate was mounted.
Binoculars and I believe Telescopes, more often use a direct image ratio 'magnification' index, so my little spotting binoculars are rates as i think 35X magnification, doesn't matter to me what the focal length is, or how many elements, all I am interested in is whether they 'get me closer' to the subject than another pair I might be interested in.
Think that they tried adopting the Binocular/Telescope magnification standard or a variant of it, for Video-Cameras, for similar reason, where the effective magnification ratio was dependent on the focal length AND the sensor frame size, that could be different in different cameras; and again, in comsumer compacts, where rather than trying to explain the focal lengths or explain the double equivilent of a 6-18mm lens on a 6x8mm sensor to the 'equivilent; focal length on a 35mm curtecy of crop factors, they just claim a 3x zoom... as an index of how much extra 'zoom' you get from the starting point, presuming that the starting point is some focal length, that on that sensor is some-where in the common 'wide angle' range, around the equivilent angle of view as maybe a 28mm on 35mm or 18mm on APS-C, the uninformed 'consumer' not THAT bothered by the technicalities, more... how much 'zoom' they get, like I am by how powerful my spotting binoculars are!
