Making a compound miter cut using a miter saw may seem difficult at first, but becomes easier once you grasp the concept, and the angle you're trying to achieve. Making a compound miter cut is simply adding another angled cut to an existing angled miter; but doing it in one cut. The most common uses for a compound miter cut are in cabinetry, hanging crown molding, and in framing, particularly the roof.
If you don't have a compound miter saw already, you need to buy one that will make all the cuts you need to make. The best, and most expensive, of these having a sliding radial arm allowing you to make cuts up to 12 inches. A general rule is to buy the best one you can afford, but buy the best one if you are planning on doing more than a couple of projects. And of course, you need a sharp blade, especially if you are doing finish work, or if you're going to get paid for the project.
Start out by placing the board, or whatever you are going to cut, on the bed of the saw. Some saws have clamps on the bed allowing you to clamp down the wider, heavier stock. Depending on what angle you need, you will first adjust the angle on the bed of the saw. Most saws start out with 0 degrees being directly in front of you, and adjustments in degrees going from your right or left, giving you nearly 180 degrees of cutting range. To make a compound miter cut, you need to find your angle, and then slide the saw into that angle using the bed adjustment in front of you.
After you have the bed angle set, adjust the angle on the arm of the saw. This arm adjustment allows the miter saw to pivot from your left or right, like a sun rising or setting, with high-noon being directly in front of you. Go ahead and make the cut. If you have a sliding arm saw, pull the running blade towards you letting the saw do the work, and then finish the cut by pushing it away from you.
Once you've made your compound miter cut, see if it fits like it should; if not, adjust the angles on your saw until you get the cut you're looking for. It will take some practice, but practice is all it takes to master compound cuts on a miter saw.