According to the American Academy of Pediatrics kids don't need extra vitamins. If you are feeding your little one a diet based on the Food Guide Pyramid then they should be getting all of the food that they need. 

However sometimes you end up with a kid who is a picky eater (such as my little boy who won't eat anything but tomato soup and bananas) and supplementation is necessary.  Some kids will stick to eating just one or two foods for months or even years so it is necessary to make sure that they are getting all of the nutrients that they need. Also if you are a vegetarian and raising your child to be one you don't have a diet that meets all of your child's nutritional needs as your child may be lacking in B12 or iron. Babies that are breastfed only may also need 200 IU of Vitamin D each day as breast milk is not fortified with Vitamin D. This is especially crucial if you live in a climate where there is not a lot of sun or if your child is confined to the indoors because of an illness.  

Older children who don’t drink at least 500ml (about 17 ounces) of Vitamin D fortified milk will also need Vitamin D supplements if they don’t get regular sunlight exposure.  Iron is very important to prevent anemia. Those most at risk of iron deficiency are infants who are not given extra iron after six months of age (usually in the form of an iron fortified infant cereal), and babies who drink low-iron formula, cow’s milk or goats milk. Good sources or iron include meats, fish, legumes, and fortified foods, such as breads and cereals.

Adolescent girls are also at risk of anemia once they begin having their periods so make sure your daughter is getting enough folic acid and iron. Your child also can't grow healthy bones with out calcium. It is an essential mineral that is necessary for healthy teeth and bones.

Kids who drink enough milk and eat enough yogurt, cheese and ice cream are never calcium deficient. Calcium supplements are poorly absorbed by children so you need to give them orange or grapefruit juice that is fortified with calcium.  Look for a juice that his high in calcium as many of them only have 20% of a daily value added.  You can also get chewable candies with calcium in them like Calcium gummy bears or Herbasaurs Calcium for Kids. Multivitamins for infants are available as drops and usually contain Vitamin A, Vitamin C, and Vitamin D. They may also have iron, thiamine, riboflavin, niacin, pyridoxine, Vitamin B12, and Vitamin E.  Good brands are Poly-Vi-Sol Drops, Gerber Vitamin Drops and Tri-Vi-Sol Drops.   

Keep in mind that many ‘complete’ multivitamins do not have all of the recommended amounts of the vitamins and minerals that your child needs each day and most don’t have enough calcium so you might have to look at supplementing your supplements depending on what brands are available.