When measuring a three-dimensional object, length, width and thickness (or height) are the three axes upon which the object extends into three-dimensional space. Distance is the measurement of a line segment between two separate points.
As defined by Dictionary.com, length is measured along the longest axis an object possesses. Width is measured as the longest dimension an object possesses at right angles to the length along the same geometric plane as the length. Height (or thickness) is the measurement of the axis connecting the highest and lowest points of an object which passes perpendicular to the geometric plane formed by the length and width.
A rectangular table, for instance, has an easily recognizable geometric plane: the table top. The longest dimension of the tabletop is the length, the shortest dimension of the tabletop is the width, and the length of any of the table legs (as they would all be equal) is the height.
Also, the dimensions of a book are an example of the measurements of length, width and thickness. The measurement of the spine is the length, the distance across the top of the book is the width, and the thickness of the pages is considered the height.