Focusing on the Problem
At the most basic level, a camera makes a record of light. When you photograph a scene, light reflected by the scene hits your camera's lens. The lens bends the light and redirects it to the recording medium. In a film camera, that medium is a strip of chemically treated plastic. In a digital camera, it's an electronic sensor.
Light from the scene bends as it passes through the lens. The light rays all converge on a single point. If you want the subject of your photograph to be in focus, you need that point to be on the surface of your film or on the digital sensor. You can adjust where the rays converge by moving the lens closer to or further from the medium. When you focus a lens in a camera, what's really happening is the lens is moving closer to or further away from the rest of the camera.
The curvature of the lens determines how far away from the lens's surface light will converge to a point. A flat lens won't make light turn in as sharp an angle as a rounded lens will. That's why the convergence point for a flat lens is further from the lens than it would be with a rounded lens. Depending upon what you want to photograph, you'll need the right lens to make sure your subject is in focus.
The standard approach to photography totals up all the light striking an image. There's no one choice with a traditional camera that can capture everything so that you can switch the focus after you've taken the photograph -- what you see in the final image is what you get. But if you could record all the individual light rays so that your image has all the visual information of a scene, you could switch from one focal point to another with the right software.
That's the foundation for the Lytro camera. It captures not just the total amount of light within a scene, but the actual light fields.