If a solid is composed of flat surfaces, such as the cube on the right, the surface area is simply the sum of the areas of the flat surfaces (called faces). So, for example, if a each edge of a cube has a length s, the area of one face is s2 since it is a square. Since by definition, a cube has six congruent faces, the total surface area is 6s2.
One way to think of surface area is to think of it as the amount of paper it would take to cover the object exactly, with none left over. So for example if the cube has a edge length of 2cm. Each face has an area of 4 square centimeters. So the total surface area is 6 times that, or 24 square centimeters. For more, see Surface area of a cube.
Although a little harder to visualize, shapes with curved surfaces have surface area too. The cylinder on the right can be thought of as two circles at each end and a strip wrapped around to form the body. (Think of it as the paper label around a soup can.)
The two circles have an area of π r2. The strip is a rectangle of width h, and a length equal to the circumference of each circle (2πr). So again we can determine the total surface area as the sum of the two circles and the rectangular strip.
See Surface area of a cylinder for an animated demonstration of this.
If you know the x,y coordinates of the vertices of a shape, there are ways to calculate the area from those coordinates. See Polygons on the coordinate plane.