First you need to do your research to determine the type of stock you want to purchase see 
Stocks: Buying, Selling & Researching  Once you have decided to buy or sell a stock you need to contact your brokerage firm (see Stock Brokers & Brokerage Accounts).  The type of account you have will affect the cost/commission/fee you will have to pay for the trade.  Also, based on the kind of account you have, this can be done either on the phone or electronically on the internet.  Again, remember you have be eighteen to trade.  However, you can set up an account with an adult.

When you place an "order", you provide your broker not only the name of the stock and the number of shares you want to purchase (or sell), but also the type of trade you want to execute (or filled).  Trade executions, in most cases are quick, but not instantaneous.  With a fast moving stock or market the price you see before you execute your trade can vary significantly from the price at which you buy or sell.  Based on the type of order you place, you can put limits or conditions on the order.  Following are the most frequent types of orders.  However there are other more complicated trades such as, "puts options", "call options" and "short sales", that only experienced traders use.  You should discuss these with your broker.