There are many, many diamond cut shapes available in engagement rings, the oldest diamond cut types are the simplest cut shapes: Emerald cut; Cushion cut; Oval and Round cut diamond being some of the first shapes devised. Over the years, these diamond cuts have been improved and modified with the advent of better technology and cutter's understanding of light refraction. Mathematical proportions have been applied to the Round Brilliant cut in particular, to enable the white light entering the diamond to split into the colours of the spectrum in the most efficient way. Of course this is what we see in the diamond as its sparkle, or scintillation. To get a great engagement ring you really need a great diamond.
The GIA ( Gemological institute of America) is a leading lab in the field of diamond grading and have spent much of its time working out the best cut parameters for the Round Brilliant Cut diamond. The very best cut round brilliant cut diamonds are given an "Excellent Cut" grade by the GIA. When it comes to cut, diamond cutters have a choice, cut the best possible parameters and get a great cut, or cut the diamond to maximise its finished weight. This is called the "yield". Every diamond is individual, just like the fingerprints on your fingers, so a cutter needs to consider where the best return will be derived from the rough diamond they are cutting. It could be that the diamond "rough" is cut into one large diamond or that in fact it will be better to cut more than one diamond from it, resulting in say two diamonds that both have a higher grade than the single diamond would have achieved. It is a complex decision based on multiple factors such as stock levels, diamond demand, yield return and grade combinations. Should the diamond be cut to retain weight or should the cutter cut away the less clean material, resulting in a smaller but better diamond? It is risk versus return when cutting. How much an engagement ring costs is directly linked to the decisions the cutter made for the diamond well before the engagement ring was even thought about.
"Fancy shape" diamonds are every cut type that is not a Round Brilliant cut, so this includes cuts such as the Cushion cut, Emerald cut, Asscher cut, Pear shape, Radiant cut, Princess cut, Baguette etc... the list goes on and on.
These fancy cut diamonds were developed in order to maximise the return from the various shapes in the diamond "rough", giving a higher yield (a heavier diamond once cutting is completed). Also, of course the development of different cut shapes allows for greater design choice in engagement rings. This often means that an engagement ring with a cushion cut, princess cut or pear shape diamond will be less costly than the equivalent weight in a Round Brilliant cut, assuming the grades are equal of course. The resulting higher return the cutter has per carat weight makes the fancy cut diamond a lower cost in the market and producing an engagement ring with a fancy shape diamond will generally be less cost than the same sort of design in the round brilliant cut.
The only snag with this is that the mix of cut parameters is infinite and what is one person thinks is a perfect cut is not so for everyone - it is a very subjective issue. This is why the GIA has decided not to nominate a set or exact cut parameter for the fancy cut diamond types. Basically it is simply fancy cut diamonds are way to varied for a definitive decision, so no cut grade is given in GIA certification.
It is for this reason that sound advice should be sought, because with no technical guidance in the cut grade you can get a very very poor looking diamond unless you know what it is that makes a great cut in a particular fancy shape diamond. You may be get the right weight and colour etc BUT a really rough looking diamond. The resulting engagement ring will reflect this. It can be why the engagement ring cost maybe really low when compared to other engagement rings that are set with better cut diamonds - you are not getting a bargain, just a really poorly cut diamond.
Many of the newer cut types are hybrid cuts combining the attributes or 2 or 3 traditional cut types. Diamond cutters often try to come up with something new so that they can get an edge in the market. This can be lucrative if they are able to brand a diamond and control the release of it by patenting the diamonds cut design. This can work for a while and give some retail outlets the control of these cut types in the engagement rings they market. If it is a really great cut then it may be copied by other cutters eventually. If it actually adds little to the existing diamond types available and is just a brand marketing exercise, then it will fade away into obscurity.
As with all diamond selection, expert advice from a professional is crucial if you want a great looking diamond.