I'm not sure which of the following is correct:
having an impact on other’s lives
having an impact on others’ lives
I just can’t figure out how the apostrophe should be used.
Since you are talking about multiple others ("...the lives of others"), then the apostrophe comes after the s.
having an impact on others' lives
If something is singular, for example, Bob. Then he is a person and we write
Bob's book.
Let us consider there are two Bobs and both of them share a single book.
The two Bobs' book
so others is plural. Consequently, the correct answer is
having an impact on others’ lives.
Let's be simple.
I have many students, and one of them is John.
I am checking exam copies.
I am checking John's paper.
I have checked all of my students' paper.
One son went to live in Australia and the other one was killed in a car crash.
One son went to live in Australia and the others were killed in a car crash.
others –> others' lives.