Visit Delta Air Lines' website (delta.com), select "Book a trip" on the main page, fill out the information requested and press "Find flights."
Search through the list of available flights and choose one by pressing the "Select" button on the right of the flight's information. On the next page, scan the flight information to be sure you've chosen the right one, and then press "Review and purchase."
Fill out the form provided on the next page. You'll need to use either a valid credit card or a PayPal account to purchase your ticket, and you'll also need to provide your name, gender, date of birth, phone number and billing address. If you have a frequent-flier number, don't forget to include that. When finished, press the "Purchase" button at the bottom of the page.