The Davy Knot is a great fly fishing knot for everyone to learn. Created by Davy Wotton, it is super simple to tie and is extremely lean and strong. I know I might have oversold the Davy Knot as being the best in the world in the title but let me say that when you are hip deep in a free flowing stream and you are trying to tie on a zebra midge to your nymph, the davy knot is easy to tie and it gets your flies back in the water instead of you trying to fumble around with making too many loops and whatnot.
I know people learn differently so I wanted to share a cool video as well as a site that has it in a step by step picture format as well as an animated gif to show you in slow motion. Pretty cool stuff. I personally like the video from Zach Matthews. He does a great job illustrating the knot and then demonstrates how well it works with small flies. He has a great tip around 1:30 mark of the video about the seating of the knot on the eye of the hook
Video on How to tie the Davy Knot
Step by step photo on How to tie the Davy Knot: