The layout square has three major functions. First and most often, it is used for making square marks on a piece of stock. They work well when one needs to make a square cut on something like a 1x4 or 2x6.
Second, a layout square has markings making it easy to determine an approximate angle of a mark or cut. Simply place the square corner of the layout square at the point where the angle meets the long axis of the stock and check the angle based on the marks on the layout square. Determining the angle of a cut is useful when trying to determine the angle for a bevel or miter on a compound miter saw or for setting the bevel on a table saw. The instruction booklet that comes with many of these squares shows precisely how use the square to determine an angle.
Finally, a layout square allows the user to mark angles in a similar manner to the method for determining an existing angle. Once again, this is covered completely in the guide which accompanies the square.
One final bit of advice: layout squares are typically available in either aluminum or plastic models. Choose the aluminum, as they're practically indestructible. The plastic ones are a bit cheaper, but they won't last nearly as long.