There are two broad schools of thought, technical analysis and fundamental analysis. Each is subdivided into many smaller schools. Fundamental analysis analyzes the "fundamentals," numeric data about a company's profitability potential as compared to its competitors. These are used to calculate an estimated "fair value" for the stock, which is compared to the current market price to determine which companies are undervalued and thus worth buying.
Technical analysis rests on the belief that the market almost instantly incorporates all available information about a company's present value and future prospects (called "discounting") through the collective activity of all investors. Technicians believe it's pointless to try to second-guess the market by analyzing fundamental data. Instead, technical analysts examine charts of past stock price and volume data for repeating patterns that they believe reveal the probability of future price moves.
Choosing Your Perspective
There is no single "right way" to choose stocks. The market is huge, and investors find success with wildly different strategies. Some investors use fundamental or technical analysis exclusively; others combine the two. It's best to have at least a passing familiarity with both, to have an informed basis to choose one or the other. Both are vast fields of study, and many books are available on each.
Fundamentals: Annual Reports
All public companies are required to disclose information on their financial state to the public. This is typically available in the form of an "annual report," among other documents, available from the company's Website. Think of an industry that you're familiar with and start reading the annual reports of companies in that sector. It's a lot to absorb, but there is an extensive literature available to help you decode the contents.
Fundamentals: Value versus Growth
Two popular schools within fundamental investing are value and growth. Value investors look for companies that are well established, with a solid track record of profits, a capable management staff, and large past "dividends" (periodic disbursements of profits to stockholders.) They believe this track record increases the chance that the company's success will continue in the future. Growth investors look for smaller companies with new ways of doing business that will disrupt an established market. These companies' stock is usually cheap compared to established companies', introducing the potential for large profits. Both methods have been profitable in the past.
Technicals: Stock Charts and Indicators
Sites like BigCharts.marketwatch.com provide free stock charts to the public. There are literally thousands of techniques for their interpretation. Popular devices include comparing the current price to one or more "moving averages," a rolling average computed each period as the average of several previous periods, and using "indicators" such as RSI or the Stochastic Oscillator, numeric values derived formulaically from past price action.