Compound Miter Saw
The compound miter saw gets its name from being able to cut a piece of wood at an angle while also cutting a bevel. In other words, the saw rotates left or right to cut the angle (usually up to 45 degrees left or right) while at the same time being able to lean over (tilt) to one side (sometimes either side) to cut the bevel (usually up to 45 degrees). The compound miter saw can be used to cut two pieces of wood so they can join together to form a 90 degree angle like in a picture or door frame but they can do much more. A good example is crown moulding in a room where two pieces meet in a corner. The crown moulding is cut at a 45 degree angle so they meet in the corner but the moulding is angled out at 45 degrees so it must be beveled at 45 degrees as well. Hence, a compound miter cut.
This example is a Dewalt DW705 Compound Miter Saw. It is an older model that is currently out of production but still a workhorse. There are many different models to choose from: double bevel (tilt left and right), sliding (the saw blade slides out and in to increase cutting capacity) and different blade sizes (again to increase cutting capacity). Whichever model you choose, select one that has more capability than you need but is still within your budget because once you start using your compound miter saw, you will find more projects for it.