To pester someone is to annoyingly nag them about something. Pestering is repetitive and bothersome.
When a child asks a parent something over and over again — like "Can I have candy? Please? Can I have candy now?" — that's an example of pestering. Parents also pester kids when they say "Clean your room" or "Take out the garbage" over and over. Pestering is like requesting, but doing it many times, usually to the point of annoyance. No one likes to be pestered.