Element is a more accurate term when referring to an individual piece of LEGO product than simply piece. The plural form is LEGO elements. The term brick(s) is often used with the same meaning but this usage is not as accurate: all bricks are elements, while there are many elements that are not bricks.
Since the company itself has sometimes used brick and element interchangeably there has been, and will likely remain, some confusion on this subject.
The LEGO Company has an element ID for each color of each element they produce. This is different from the Design ID which generally corresponds to the part number assigned by BrickLink, Peeron and others.