So firstly, 'affect' is a verb: 'to affect' and is different from 'effect', a noun, so when we see 'effect' it is the 'impact' on something.
Let's use an example:
"What are the effects of drink driving?
Driving concentration has been affected as well as reaction time"
I just used the third word in your list which is used for two meanings : like above for effect and also a physical impact, when an aeroplane( airplane in US spelling) hits the ground it makes an impact.
Finally, influence is a verb and a noun, we use this term in a more metaphorical (from metaphor implying something different, non-literal) or cultural sense.
An example of this is "people in the Renaissance were influenced by ideas of the ancient Greek and Roman thinkers and artists".