I started out with this square chunk cropped from my logo:

In Photoshop, I saved it as a 16-pixel square image and resampled the image using the bicubic sharpener (this keeps the image from blurring as it is resized). Here’s how it looks blown up:

And here’s how it looks at actual size:

Cute! You can see how it looks in a web browser by going to my website, www.bluemoonriver.com
Have a website and want to make your own favicon? Here’s a great tutorial by Jennifer Apple on how to make one in Photoshop.