Kids and teens can own stock by having an adult on the account with them. This kind of account is called a custodial account. The minor (referred to as the beneficiary) owns the stock, while the adult (the custodian) has legal responsibility over the account until the minor turns 18.
The minor can set up his own log-in (separate from the adult’s) so he can see how his stocks are doing anytime on his own. In addition, young people can place stock trades with the adult’s consent. For example, if a 15 year old wants to buy $20 of Facebook stock, he sets up the transaction, which is routed by email to the custodian (say, mom) for approval. If mom approves, an order to purchase $20 of Facebook stock is placed and sent to the market to execute. Otherwise, no order is placed.