It may help to think about post processing in these terms:

When shooting film, you either took the film to the lab, or developed the film yourself. During this "process", you had the opportunity to further tweak the image to produce the desired effect. In the days of film, it may have been the case that you would have made several test prints until you reached a result that you were happy with.

When shooting a digital camera, your camera automatically does "post-process" to produce a JPEG image. The camera has a tiny little computer that does a few specialized calculations on the information captured by the camera's sensor, and produces a "ready-to-print" image. Many cameras offer settings which will allow you to control the "final" output, but ultimately the camera is doing post processing.

So to your question: is post-processing required. Yes. Otherwise you would have meaningless information (a film negative or raw binary data), neither of which would make for a compelling image.

Now to your question which I think is more "Is it required to edit photos on the computer" for best results. That answer is depends. If you look at my example about the camera's little computer creating the image, think of post-processing with a computer and your brain, as like processing your images with like a NASA super computer. Instead having a little computer constrained by it's own designs, you have a human with all it's strengths and flaws, creativity, life experiences, knowledge, etc manipulating the image. Clearly the human computer has more capacity to create something more than just a mere visual representation of a bunch of sensor data. But that same human can totally screw things up, producing an image that is terrible compared to what was done by in-camera processing.

That said, I believe getting the best out of your images is a two-phased approach:

1. Taking the best picture possible with your camera. Nail the exposure, frame the subject properly, etc.
2. Post-processing to enhance your image.

To the first point: No amount of post processing can fix a poorly shot image. The closer your initial image is to "perfection" the less time you will need to correct flaws in your image, and the more time you will have to make the tweaks to finish off your image.

To the second point: there are limits to what you can do with your camera. You may not be always be able to frame your image in a way that is compelling, and removes that unsightly powerline. Or, perhaps you didn't notice the piece of trash that ruins an otherwise lovely shot of your daughter at the park. Whatever the case, the post-processing step can fix these issues, and add the proper enhancements to your photo.

Me personally: Adobe Lightroom 3 is an integral part of my workflow. I use it to convert my RAW images, and more often than not, apply various settings to each image. However, when I shoot the image, I take care to try and get the best "negative" I can. Additionally I have a mental picture of what I'd like the final output image to look like, and I use tools like Lightroom and Photoshop to achieve that final image.