The most common place you find a straightedge and compass being used is for math, or more specifically, geometry. Using a straightedge and compass, one can create perfect circles and perfect lines to aid in calculating things such as angles, radii and the area of a circle.
Architects often use a compass and straightedge to aid in drafting their designs. While the straightedge can help them draw straight lines of exact lengths, the compass can be used to add arcs and circular constructions to drawings. Since the compass and straightedge can be used to create lines and arcs of exact lengths, architects can more accurately determine real-world proportions for their scaled drafts.
Historically, straightedge and compasses have also been used for navigation. Using a map, straightedge and compass, navigators can calculate distances from different landmarks. The straightedge is used to plot lines from one point to another on a map while the two legs of the compass are spread from one point to other, then applied to the maps' scaled distances to figure out the distance in nautical miles. This is how navigators figured out distances before the advent of the magnetic compass and modern GPS equipment.
Similar to how they are used in architectural drafting, a straightedge and compass can also be applied to the creation of art. Creating straight lines, circles and arcs can be applied to an artist's drawing. The straightedge can be used to create straight lines to form the base of an artists drawing, such as horizon and vanishing points. The compass can provide the base for circular objects, such as a person's face and eyes.