A piston rod is a linking component between the crankshaft and a piston. It is bolted together around the crankshaft and usually has at least one passageway for oil to pass through. It is connected to the piston by a hollow pin which passes through both components.
In some engines (usually in go-karts and lawn mower engines, not cars), there is an extension which allows the motion of the rod and crankshaft combined to splash oil on the internal components (especially useful in engines without oil pumps).
Replacing a piston rod with a longer or shorter rod will have no effect on the length of the stroke (the distance the piston travels up and down), but it will have an affect the location of the piston at the top and bottom of each stroke, which will also alter the compression ratio of the engine. High compression engines sometimes require stronger piston rods.
Putting in lighter-weight piston rods will help it change rev speeds more quickly, and slightly improve the power output. They are especially beneficial in engines with a high peak RPM.