Learning the Axel is the first real big hurdle all figure skaters (freestylers) face. The transition from half jumps to one-revolution jumps usually comes rather easily. Multiple revolution jumps are infinitely harder. It is like, on the ground, just about anyone can jump and turn around... but to jump and turn 2 rounds is a totally different story!
The Axel is also intimidating because of its forward take-off. It is sometimes said that the Axel is merely a Waltz jump with one more rotation. However, over the years of learning the Axel and teaching it, I find that it is not just an upgraded Waltz.
A good (beautiful) Waltz jump will have extension in the air. This extension causes a delayed shift of weight on the landing foot. Being just a half-revolution jump, this is not too difficult to achieve. Such extension in an Axel will give a delayed Axel. While a well executed delayed Axel is beautiful, it is (trust me) so much harder to do, and is not what skaters beginning on the Axel should focus on!
So, what does the Axel, need? A quick knee action on the jump, and a quick snap onto the landing side while straightening the landing foot. One of my former coaches said, think of stepping up a stair. Though this is not the only thing I think of in an Axel, I dare say it is one of the most important!