This is basic information concerning what you should be looking for in running shoes. You will need a good pair of running shoes. Running shoes are made to cushion and support the foot in ways regular sneakers are not. It should be noted that good running shoes should not be used for other sports. They are not made to support the foot for sports, such as basketball, which require lateral motion. I have a personal preference for Brooks running shoes, although many companies make good models.
Read this paragraph carefully! Most of the major shoe companies make good quality running shoes. You have to realize that running shoes are made for this sport only, and cross trainers and other sneakers are not appropriate for running. The major characteristics of running shoes are cushioning, stability, motion control and flexibility. When you look at shoes, make sure they flex easily in the forefoot area, and not in the middle of the shoe. If they do not flex in the forefoot, and rather flex in the middle, you will end up with calf and Achilles tendon injuries. Make sure they have that company's cushioning system (Brooks Hydroflow, Nike Air, Asics Gel, etc.) The cheaper shoes made by each company do not contain the cushioning elements. With regard to stability, if you hold the front of the right shoe steady with your left hand and twist the shoe at the heel towards the arch area or inside of the shoe, it should not flex in that direction too easily or there is not enough support.
2013 Update: Many running shoe companies now make minimalist shoes to take advantage of a market created after the publication of Born to Run by Christopher McDougall. This book advocates for a barefoot (or close to it) running model. I know many athletes who have done well with this model. It seems that the model is better suited for grass and well cushioned venues. The model does not do as well on streets of macadam and concrete. A good running store can help make sure your child/athlete gets an appropriate shoe.
The best way to ensure that you get an appropriate shoe and an appropriate fit is to purchase your shoes at a reputable running store (not some place like Sports Authority which charges top dollar and the people have little knowledge about running).
Some running shoe stores to consider:
Morristown Running Company: CLICK HERE
Runners Haven in Randolph: CLICK HERE
Ridgewood Running Company: CLICK HERE
Road Runner Sports. A national mail order company with selected stores around the USA has a store in Paramus, NJ: CLICK HERE
Super Runners Shop. In New York City and Long Island, including the NY Armory Track Center: CLICK HERE Included for parents who work in New York and for nostalgia. I ran for Super Runners back in the late 1970's and early 1980's.
