Things that seem harmless to adults may be scary for children. We as adults can help our children over come their fears by show patience, understanding and listen to our child.
 

Babies (8 to 12 months old) 
Since they can't talk about these fears, you should try to eliminate them as much as possible: They are often frightened by everyday things may include the dark or loud noises like thunder or the vacuum cleaner.  Use a night light; or run the vacuum cleaner when they are not around. You can't stop thunder, but you can be there to reassure your child that everything's OK.

Babies may also experience separation anxiety, which usually peaks around 10 to 18 months, though it can happen even earlier. At this age, they begin to distinguish between familiar and unfamiliar situations, which may cause them to become fearful of strangers. 

Toddlers and preschoolers (2 to 4 years old) 
It's normal for toddlers and preschoolers to have fears. At this age, children have vivid imaginations, and have difficulty distinguishing reality from fantasy. They may also find new things to be scary. 

Everyday situations may frighten toddlers, such as bedtime, bath time, or going to the doctor. If the adult shows nervousness, the child will notice this and may become nervous themselves for reasons they don't understand.

It's important for parents to take their child's fears seriously because they are real to them. It may be "something under the bed" to maybe a barking bouncing dog.

At the age of 3 or about, this is when there is little crying and fear when separating from a parent. If the parent makes a big thing about it, the harder for the child to accept that the parent is leaving. Always reassure them the parent is coming back.

Little ones have bad nightmares at night. Talk to them, have them understand the difference between their dream and real life. Stay with them until they sleep.

Children who experience a night terror (not the same as nightmares) may wake up screaming and thrashing, but they are only partially awake and won't necessarily be aware of your presence. They will not respond to you, and will usually fall back asleep without completely waking up. They won't remember it the next day. 

School-aged children (5 years and up) 
Fears of school-aged children tend to be more reality based, such as storms, fires or injury. But the fear may be out of proportion to the likelihood of these things happening. As their understanding matures, these fears generally go away. If they seem upset by you and your spouse arguing - reassure them that its not about them and that sometimes adults need to argue, that you are not separating.

Being exposed to media may also create fears in young children. Images from movies, video games, music videos, Internet web sites, and even television news stories can make children afraid. 

Unlike young children, older children may express their fears in ways other than crying. They may bite their nails, tremble, or suck their thumb. They won't necessarily tell you they are afraid, so watch for signs.

Tips for Parents
• Never force your child to face the fear before they are ready. Allow them to work through it at their own pace. When they do, be sure to give lots of praise. 
• Respect that the fear is real for your child. Don't belittle your toddler or make fun of them. 
• Help your child work through the fear. Read books, make up stories or act out situations that deal with your child's particular fear. Tell them how you felt when you were their age.
• Try to desensitize your child to the fearful object or situation. Using a toy fire engine may help to reduce the fear of the real one. 
• Help your child feel physically secure by hugging them, holding their hand, and being close. You can also teach them to take long, deep breaths to reduce their anxiety. 
• Encourage your child to share their fears with a “worry doll”—either use one of their existing dolls or make one together. 
• Try not to reinforce the fear by being scared yourself. Overprotection can also cause children to be unnecessarily fearful. 

Limit your child's exposure to media that may create fears or make them even worse. This includes TV, movies, video games, Internet, and even printed materials. You can also help teach children good media habits, which will help them distinguish between what's real and what's not real. 

Source/Reference: 
 
 

This information should not be used as a substitute for the medical care and advice of your physician. There may be variations in treatment that your physician may recommend based on individual facts and circumstances.