To begin understanding diamond cuts, first, you have to release your preconceived idea of what a diamond cut is. I know the sorts of words you've got in mind, things like "princess" and "emerald" are coming to mind, right? Those actually refer to the shape your diamond has been made into, not its cut. 
"Cut" is a much more technical term which communicates a diamond's capacity to shine. Cut is what creates glitter and sparkle. Cut is also what separates good diamonds from great diamonds, literally. Knowing cut types and what they mean and what you ought to be looking for can save you money and prevent you from overpaying for a lesser quality diamond. 

In a very basic sense, you should be looking out for three different types of diamonds. Shallow cut diamonds are those which allow for light to sink to the bottom. These appear dark because of this, even if they are large. Moreover, if a diamond is cut too deep, light runes through the sides, leaving a dark, dull diamond. What you want is the sweet spot between the two, the kind that catches the light no matter the direction you turn it. 

Different diamond appraisers will assign different names to the cuts of diamonds, making it rather difficult to assess yourself unless you are up to date with what each means when they say one thing or another. For instance, most assessors go by a grade system which descends from Ideal to Premium to Very Good to Good to Fair. However, what is "Ideal" to one company is actually "Excellent" to the Gemological Institute of America. Others, still, alter the rating systems according to their own business, assigning their own name to the best cuts. And, these grading systems are left generic enough to leave one confused if one has not done the research to understand the system. 

Keep in mind as well that different shapes of diamonds will alter the assessment of the diamond, so, again, think about that when listening to a sales pitch. What is very good for one shape may be good for another shape.

You may be wondering what types of factors go into determining the assessment of the cut. One major aspect are the number of facets on a certain diamond. A facet is a flat surface on the diamond, usually triangle shaped.

You may think that the more facets a diamond has, the more spectacular it must be. However, it is a common practice to obscure diamond defects with additional facets. Therefore, if you are looking at a round diamond, search out one that has near about 33 facets on its upper portion, known as the crown, and around 25 on the lower section, which is also called the pavilion. The line of differentiation between them is called the girdle. 

Finally, always remember to check your perfect cut diamond against its color, carat, and clarity. All four play an important role in determining the value of a diamond. A little time pre-researching what you want and what falls into your budget will help more than anything else in getting the most for your money.