Why isn't my 1995 Honda Civic starting?

I have a 95 Honda Civic that only starts SOMETIMES. I noticed there is a subtle clicking noise that NEEDS TO HAPPEN before I turn the car on and it does this inconsistently. It's not the battery since I just replaced it. In order for me to turn the car on I have to turn the key back and forth until that clicking noise happens then it turns on. What is it?

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