Diamond shape can have a significant influence on the overall aesthetic of your engagement ring. While there are others, the most common options for diamond shape in engagement rings are the following: round or round brilliant, princess cut, asscher cut, emerald cut, cushion cut, radiant cut, pear, oval, marquise, and heart.

Contrary to belief, the wholesale pricing structure used by dealers to negotiate on diamond prices does not take into account diamond shape other than round or fancy. Round represents modern round brilliant cut diamonds. Fancy represents all non-round brilliant diamond shapes. There is a premium paid for round brilliant diamonds.
Diamond price example.
For example, if we assume we’re looking at two diamonds, each GIA certified, each weighing 1.50 carats, with F color and VS2 clarity and both with ideal makes, no fluorescence, et al, we’ll find out by examining diamond dealer wholesale lists that the stones will have different prices. The industry high cash asking value for the round brilliant diamond is $16,800 the non round brilliant diamond (say princess cut for argument sake) would be $12,600. Pretty substantial price difference all things considered.
Now market trends will also influence the type of discount you might receive on a given shape, but that influence is not due to different pricing structures. It’s just due to the consumer demand. Marquise shaped diamonds, pear shaped diamonds and heart shaped diamonds at or under the 1.50 carat mark are not hugely popular currently. This means that there is probably more room for discount on such stone from the wholesale list b/c dealers would rather exercise that type of inventory, turn it into cash, and stock something more likely to sell.
As mentioned earlier, diamond shape will have a substantial impact on the overall aesthetic of your engagement ring as well as how the ring “fits” on the hand that’s wearing it. Elongated diamonds, like ovals, marquise, emerald, traditional radiants and even some cushions fit well on longer fingers in many jewelers’ opinions. These elongated shaped diamonds will also impact components of your engagement ring, such as head type, shank width and complementing side stones. Mounting a step cut stone like and asscher or emerald as your showcase and flanking it with high faceted round brilliants or trillions of any moderate size can produce a look that might not be the most desirable. Mounting high faceted with high faceted accent stones, or step cut with step cut is usually more common b/c doing so will keep a uniform look and prevent that which is meant to compliment the showcase diamond from taking away.
These are just some aspects of diamond shape.