There are a number of reasons why lights might flicker.
Incandescent bulbs running on AC will, of course have a cyclical input although the fact that the filament stays hot tends to smooth the actual light output out a lot. For similar reasons fluorescent tubes will also flicker at the mains frequency and this tends to be rather more pronounced.
LED lights of course run on DC power but most are controlled by pulse width modulation as this tends to be the most efficient way to provide a regulated current supply and dimming.
How this interacts with cameras can be a bit complex and has to do not just with the frequency of the lights but the sampling rate of the camera and the phase angle between them.
For example if you have a situation where they camera and lights have the same frequency the light level can seem (to the camera) brighter or darker depending on the phase, ie if the camera is recording at 40 frames per second and the lights are flickering at 40Hz then depending on exactly when you start sampling you could be consistently in the dark or light stage of the cycle or somewhere in between.
Obviously it is very difficult to measure or control this phase difference whcih can make consistent exposure difficult.
Equally if the frequencies are different you can get a variation in light levels very similar to the wagon wheel effect. https://en.wikipedia.org/wiki/Wagon-wheel_effect
You can also get problems with still cameras on short exposures.
Clearly lights designed for use with film and video tend to be of a type with little or no flicker and for this reason LED lights are much more common in theatre than film.