One common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations.
The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, etc.) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together.
Because the region filled by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured.
This error can be reduced by dividing up the region more finely, using smaller and smaller shapes.
As the shapes get smaller and smaller, the sum approaches the Riemann integral.