There are as many ways to cut down a tree as there are loggers, but this is the safest way to do so.
The procedure has three points of interrest:
The notch, the hinge and the back cut.
The notch is made with two cuts, the first one being the top one at an angle of approximately 30-45 degrees.
Stand with your left shoulder leaning against the tree, holding your chainsaw against the trunk. The direction the tree will drop in is determined by the angle of this first cut. Most chainsaws have the front handle at a 90 degree angle to the bar, so use the handle to aim.
Make sure you hold the bar horizontal as you start cutting into the tree.
The next cut is made from the same position, also making sure the bar is horizontal. Look through the first cut to see when the two cuts meet. It is important they dont overlap, as this will weaken the hinge.
When you are done, inspect the notch for overlaps and size. The size should be 1 third of the width of the trunk seen from the front of the notch. The bottom surface should be level and flat so the tree doesn't change direction mid drop.
Next, decide on the thickness of the hinge. The bigger the tree, the bigger the hinge should be to ensure a controlled drop. 1 to 2 inches is normal.
If you decide on 1 inch, you start the back cut 1 inch behind and 1 inch above the hinge. Insert the tip of the bar and drill horizontally into the trunk looking at the bar and into the notch to make sure your direction is correct.
The bar should follow the front of the hinge in order to ensure the hinge is equally thick all the way. Once the bar is as far in as it will go, check again that it is parralel to the front of the notch.
Then start cutting back through the trunk but not all the way. Leave a bit at the back of the trunk to hold the tree untill you are ready to drop it.
Now go to the other side of the tree and repeat the procedure.
The tree is now standing on the hinge and the bit at the back, and should still be safe and stable. If there is no doubt that wind and weight will make the tree drop in the desired direction simply cut the bit at the back away slightly below the back cut level.
You cut approximately at the same level as the bottom of the notch, and the tree will split the fibres and fall without throwing your saw away.
If however there is a chance the tree will fall backwards or not at all, you insert a felling lever as far back in the back cut as possible, next to the bit you left. Now, as you cut under the bit, the felling lever keeps the tree standing untill you apply pressure.
If the tree refuses to budge, apply wedges next to the felling lever.
That is how to cut down a tree in a controlled manner. Most forest workers build up the skill to judge a tree and skip some of these steps, but till you get to that level this method will quarantee that you have full control at all times.