Cut is what determines how well-cut your diamond is, where as diamond shape is the shape it was cut into.
Do not confuse with Round. The Round Brilliant is the modern version of the Round which has been refined for maximum shine. The Round Brilliant is by far the most popular and has the best angles to shine maximum brilliance.
A square cut diamond that has refractive properties almost near round brilliant. The princess is the preferred square cut shape over Radiant and Asscher.
A more traditional shape, the Emerald is not as popular as it once was, but still retains its old world elegance.
Popular before the Princess shape was around, the Radiant has more facets than a Princess, but has the corners trimmed like the Emerald shape. This shape is not widely popular.
Hard to find due to low demand, but some people prefer a Heart shape diamond for sentimental purposes.
Like the Emerald, the Marquise is a traditional shape. This is probably the fourth most popular shape behind the Round, Princess and Oval.
Mostly used in pendants, the Pear shape diamond is shaped in a tear drop shape and has fairly good proportions to refract light well.
A square cut similar to an emerald with step facets; however, it is more square rather then rectangular shaped. Like the emerald, it has clipped corners. Recently the Asscher shape has become very popular.
Not as popular for solitaires, but very popular for three stone anniversary rings, with two matching diamonds on the sides.