What is crop factor and how does it relate to focal length?

The above answers are correct but they do not point out that the focal length is exactly the same. The conventions in photography sometimes makes a machine vision dude like myself want to pull my hairs out 🙂
FOV = 2*atan(size/(2*f))

size is the size of the actual chip. You compute it for height and width separately, like 36mm and 24mm (for full frame) and 25.1mm and 16.7mm for your std issue “crop camera”, or 4.8 x 3.6mm for your std issue 1/3 inch machine vision camera with c-mount.

If you start to linguistically code it as “having a longer focal length” then you might believe that it gives a larger magnification, which it doesn’t. I also noticed that the joyful world of photography even try to fix that by introducing a virtual “35mm eq. magnification”, which makes no sense either, since M is based on the physical size of of the projection and not dependent on sensor size at all.

A third term to be vary about is circle of confusion which is about how focused the light rays are through the lens onto the sensor. You will find calculators that calculates the the lowest COC (e.g. for depth of field) based on what can be detected by the human eye as a point. I’m not going to look at the projection through the lens on a wall, am I 🙂
If I look at a zoomed in digital photo on the screen, or an algorithm process machine vision VGA image I want it to be sharp within a pixel cell size (e.g. 6um) and not some print human based measure that will never apply to the images I take. And then the depth of field suddenly becomes much narrower than those calculators show, as they consider the coc limit to be 29um for full frame and 18um for aps-c.

So in conclusion, you need to keep the terms seperate. “crop sensors” affect the FOV (because you change 1 out of 2 factors in the formula), not the focal length. Since focal length affects more than FOV, you cannot convert the focal length.