Some simple illustrations of these concepts:
Cut: as we will see soon, the cut of a gem must be controlled by the characteristics of the gem (refractive index). If the cut has the wrong proportions, it will look dull. This can be seen when glass is cut to resemble diamond
Clarity: a gem's value is reduced significantly by the presence of inclusions. An example shown here illustrates the effect of inclusions in diamond.
Color: ideally colorless, except 'fancy' diamonds. Color is graded in corrected daylight by comparison with standards. Grades range from colorless to light yellow. Note, however, that exotic colored diamonds exist, and these may be quite valuable (e.g., "yellow, pink, blue, green, brown, black, etc!).
Carat weight: one carat = 100 'points'. Weight is extremely important (as it determines size) in determining value, but may be outweighed by the other factors listed above.