While the specific steps vary among companies, you usually need to make an irrevocable election to exercise. You do this by completing an option-exercise form (on paper and/or online) before the expiration of the option term. The form is sent to the specific contacts at your company and/or the third-party stock-plan administrator. Sometimes different forms are used depending on whether the exercise is by cash or is cashless.
For grants in a public company, you also need to establish an account with a stockbroker, even if you are not selling shares in a cashless same-day sale or a sell-to-cover exercise. The broker will also have its own forms and IRS forms you need to complete. Typically, your company will have selected one or more "captive" brokers for you to use. You may need to clear any stock trades at exercise with your company's legal or compliance staff and provide additional authorization to the broker to execute the stock sale.
Alert: If you are uncertain about the procedures, rules, and deadlines, ask the appropriate person at your company to explain. Companies follow their rules for stock options very precisely.