If you need to cut hard wood, a good axe will be your best friend. A saw is great for wood workers and for precision art; however, felling a tree and splitting it into boards is only a job for an axe.
Decide what you need to do with an axe. Choosing an axe has everything to do with your purpose. Axes can be used for either cutting or splitting.
Look at the many axes out there. If you want to cut wood and fell (cut down) trees, you need a cutting axe. A cutting axe has a very thin blade that comes to a very sharp edge. This sharp edge can be easily sharpened without taking too much metal off. Make sure the head is made of good solid steel, and is not smooth with a dent here and there. Look carefully at the head; it needs to be one piece of metal. You should not see any joints or connections.
Pick up the axe and look down the handle to the head. You want the handle to be wood and the grain (the lines in the wood) should run toward the head. The length of the shaft also depends on your purpose. To fell trees, a handle at least as long as the distance from the middle of your palm to your arm pit will be needed. If you will use it only for small or close cutting jobs, the size of your forearm will be fine. The handle should not move in the head, and when you place the axe an inch below the head on two fingers, it should balance.
Look at the other axes out there. If you need to split wood, you're going to need an axe that will be able to handle this process or you will literally break it. A splitting axe will be made of more steel than a cutter, and the blade will be sharp and very wide and look like it could truly move two pieces of wood apart from each other. As with the other axe, make sure that the axe head is made of one piece of metal.
Lift the axe you want to buy and rest the head on your shoulder. If this process takes a lot of effort, you will not be able to swing the axe.