It's normal for children to fear something at one time or another. Babies and toddlers experience fear of strangers and begin clinging to parents. They also experience separation anxiety when one or both parents are not around. Our fifteen month old often demonstrates this behaviour. If somebody she doesn't recognize or feel comfortable with comes over to our house she hides behind me and clings to my leg. Sometimes she cries if family members hold her. Our four year old used to be like that. She used to have a fear of being around a large group of people, but she grew out of that. She has a normal fear of the dark. She likes to sleep with a little light when she goes to bed. Sometimes a fear is brought on by a traumatic experience. Our four year old and five year old never used to be afraid of flies. One day we went to visit my in-laws up north and they were bit several times on the neck by black flies, which caused them to bleed. Now they are terribly afraid of flies.
Children may start being afraid of something because another adult in the family has exhibited that same fear. I have an extreme fear of spiders, but I try not to overreact in front of our children.
You can help your children overcome their fears by talking about them. It's important to be supportive and sympathetic and not belittle them. You can help them eventually overcome their fears by encouraging them to confront them.