It varies so much from one school to another, I don't think there's such thing as a generic job description.
At my current school (an approved FTO), the CFI does all the 170A flights for CPL and IR students if he's available. He frequently flies with students who need a little more brushing up after their 170A. He keeps up to speed with the progress of every student, and gets involved in their training when necessary. He sometimes does other work which takes him away from the school, such as examining for other schools.
In my previous school (a RF), the CFI was very rarely around, and I reported to the flying school Manager on a daily basis (who was not an instructor). The CFI did a checkout with me when I first started working at the school, but apart from that he was flying or managing the AOC side of the company far more than the school. About the only time we saw him in the school was when he was teaching his 17-year-old son to fly (although he delegated that to other instructors soon after his son went solo). Although he is a great bloke, and a very good pilot, his involvement in the school was really summed up by his son, who said "If my dad's the CFI, how come he's so out of touch that he didn't realise I was supposed to do the Air Law exam before I went solo?"
Two different extremes, and although neither of them are formal job descriptions they should give you an idea of the range of duties.
FFF
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