Go to one of the aggregation sites which offer price comparison, or allow you to set your own prices, such as Priceline or Orbitz.
Select the "Flights" tab. Enter departure and arrival cities in the "From" and "To" boxes. Typing the first letters will give a list of available cities to select from. Enter the dates for the flights in the "Departing" and "Returning" boxes, or using the built-in calendar.
Fill in the number and type of travelers. Click the "Search" button. This may have different names depending on the site.
Verify the prices on the resulting page. Select the flights for departure and arrival. Prices presented are still an indication; it's not until you order the tickets that you actually see their cost.
Fill in the traveler details, verify the flight details and personal information. Click the "Buy" button (which may have different name on different websites). The final ticket price appears.
Cancel the process to compare other prices, or confirm the purchase. Fill in the payment details and confirm the order. The tickets will be delivered electronically.