You don't get a direct say in a company's day-to-day operations, but, depending on whether you own voting or non-voting stock, you may have a hand in shaping its board of directors and deciding on special issues.
Voting Stock: If you own voting stock and you're a shareholder of record when a decision must be made through a vote, you have a right to vote on it. The right to vote for a company's directors or on a specific business decision is similar to the right to vote for an elected government office -- e.g. senator or local mayor -- or on an issue put to a ballot vote. As a registered voter, you aren't obligated to vote. As well, you don't really get a direct say in daily government operations (although you do vote on the people that do). The one main difference between voting as a citizen and voting as a shareholder is that if, as a shareholder, you choose not to submit your vote, there is the possibility that a default choice will be made regardless of your true desires. That's why it's important to carefully read the fine print on the proxy form sent to you.
Non-Voting Stock: A non-voting stock doesn't allow you to participate in votes affecting shareholders and the company. With this class of shares, investors forfeit their right to have a say in the direction of the company for what is often an incremental stock price advantage over voting shares.
For example, Alphabet Inc. (the company formerly known as Google) in April 2014 split its stock into three classes of shares to preserve the control of founders Larry Page and Sergey Brin. Class A (GOOGL) shares are held by regular investors with regular voting rights. Class B shares (which don't typically trade in the open market) are held by the founders and have 10 times the voting power of the Class A stock. Class C (GOOG) shares have no voting rights and are normally held by employees and Class A stockholders. (See: What's the Difference Between Alphabet's GOOG and GOOGL?)
While the two classes of stock have historically tracked each other in price movement, those voting rights will cost you more, as GOOGL shares typically trade at a premium to the GOOG shares. For example, on Dec. 12, 2017, GOOGL shares closed at $1,048.77 a piece, while non-voting GOOG stock closed at $1,040.48.
Not all companies offer these two different types of stock, and not all types of voting stock have the same voting rights. If you are interested in playing a part (albeit a very small one) in the decision making processes of a company, make sure you buy the right type of stock.
If you would like to learn more about stocks and the stock market, check out this Stock Basics Tutorial. For more on the issue of shareholder rights, check out this Knowing Your Rights As A Shareholder and What is the investor rights movement?