A compound miter saw is designed to make precise bevel (along the Y-axis) and miter (along the Z-axis) cuts, either independently or simultaneously. However, if your miter saw has been used on a job site, meaning that it has been bumped or banged around getting it into and out of a truck, or any one of a hundred other job site hazards, it is probably at least minimally out of alignment.
If you really want to get down to it, most miter saws are at least somewhat out of alignment the moment you take them out of the packing crate. While most alignment issues won't affect the safety of the miter saw, the certainly can affect the accuracy, which can become particularly evident when performing such tasks as matching corners with miter joints.
Clean Your Saw
The first place to begin with tuning up and calibrating your miter saw is to give it a thorough cleaning. Use an air compressor with an air nozzle on the end of the hose to blow out sawdust out of every nook and cranny of the saw and table. Then wipe down the entire unit with a clean, cotton cloth so you are working from a fresh starting point. As you're wiping it down, check the entire saw for pitch build-up, particularly if you cut a lot of softwoods with your saw. If you find excessive pitch build-up, you can use cleaners such as Boeshield's Rust and Stain Remover to help reduce the pitch build-up on the saw.
Check the Blade
Before getting into calibrating your saw, take a good look at the saw blade. Not only should your blade be sharp and clean, but it also should be appropriate for the type of cutting you'll be doing. There are many different sizes and styles of miter saw blades from which to choose, but for most projects, a good combination blade of the 60-72 tooth variety performs well for a versatile number of tasks.
Most modern saws have very sturdy and flat table bases from which to work, ones that don't require an adjustment. If your saw is much older, you may need to adjust the table to ensure that it is flat. One way to check the table for flat is to place a metal straight-edge or the long edge of a framing square on edge along the top of the table. You can merely eyeball between the straight-edge and the table to check for any gaps or slide a thin piece of paper between the two for verification.
Once you know the table is solid and flat, the first thing to check is that the fence along the back edge of the table is straight. Once again, a metal straight-edge or framing square can be placed along the length of the fence to check for any spots that are out of alignment. Most miter saws have a split fence, with one-half to the left and the other half to the right of the blade, and each section should have at least two set screws or knobs for loosening and tightening the mounting of the fence. Loosen and adjust each side of the fence until both are straight and in line with one another.
Once the fence is straight and true, you now have a solid point from which to check and/or set the miter angle of the saw. Begin by setting the bevel angle of the blade to 0-degrees (straight up-and-down), and adjust the miter angle so that it is also at 0-degrees, which should be square to the fence. Unplug the saw from the power outlet for safety, then place a rafter layout square or combination square flat on the table, with one edge against the front edge of the fence. Next, lower the saw until the blade in the lowest position possible, and slide the square against the side edge of the blade (you may need to manually raise the blade guard to have access to the side of the blade).
With one edge of the square securely against the fence and the other edge against the side of the blade, check for any uneven gaps between the fence and the blade. If you find any, loosen the miter tightening knob and adjust the miter angle until the blade is as square as possible to the fence, and tighten the knob to hold the angle. You can then loosen the screws on the detent plate and adjust it accordingly so that your saw's zero-degree detent should always be accurate.
Some compound miter saws can bevel up to 45-degrees to both the left and the right, while others can bevel only left. In either case, if you set up the 0-degree angle to be square to the table, all other angles should be precise as well.
To check the bevel angle, place your square on edge, with one edge flat on the table, and the other edge rising straight up from the table. Lower the saw (with the blade guard raised) and slide the square until the vertically-oriented edge of the square is aligned with the blade. Visually (or with a piece of paper as discussed earlier) check for any gaps between the blade and the square. If any are present, loosen the bevel adjustment knob and adjust the bevel until the saw blade is aligned with the square. Tighten the knob to hold the bevel angle, then adjust the detents for the bevel. The method of doing so will depend on your saw model, so consult your operation manual for details.
Check for Accuracy
Once you have completed all of the adjustments, it's time for some real-world testing. Plug in your miter saw to the power outlet and place a flat, wide board (such as a 1x8) on the table against the fence, and make a crosscut at 0-degrees miter, 0-degrees bevel. Remove the board from the saw and check both angles with your square for accuracy. If the cuts aren't square, you may need to re-adjust the saw.
NOTE: This test is far better suited to checking the miter angle than the bevel angle. For a better verification of the bevel angle, place a 2x4 on edge flat against the fence and table and make a 0-degrees miter, 0-degrees bevel cut. Then check the angle of the bevel with your square, and make any necessary adjustments to the bevel detents based on your findings.