Diamonds are graded by the "4 Cs" Colour, Clarity, Cut, Carat - Generally by the GIA system, IDC (International Diamond Council) Rules or the CIBJO Rules, they are all very similar systems with just slight variation in the names for some of the colour and clarity grades.
There are many labs that grade diamonds using one of the systems mentioned above.
Within the diamond trade GIA is the highest regarded laboratory worldwide, followed by HRD and IGI, there are many other labs some are very lenient with their grades such as EGL and others are regarded as rather harsh.
If you are absolutely set on a Certified Diamond we would recommend choosing a stone with either a GIA or HRD certification.

There is no magic involved in grading diamonds, it is done by an individual by eye and ultimately graded using a 10x magnification lens and is only the opinion of that particular grader on that day.
There are many examples of diamonds being graded by a lab as one grade, then being sent back at another time to the same lab and coming back with a different grade.
But it is still a good guide and reference for you if you feel that you need that.

But our main advice is always buy the “diamond” not the “paper”

COLOUR

The diamond colour grading scale varies from totally colourless to light yellow. The differences between one diamond colour grade and another are very subtle and difficult to distinguish
D-F : Colourless: perfect or almost perfect colour.
G-J : Near colourless: good to very good colour, and this diamond may "face up" colourless when mounted, especially in white gold or platinum.
K-N : Feint but noticeable yellowish tint.
M-Z : Tinted Yellow


CLARITY

Diamond Clarity is a measure of a diamond's internal flaws and impurities. A diamond that is virtually free of interior or exterior inclusions (commonly referred to as flaws) is of the highest quality and nothing interferes with the passage of light through the diamond. To determine a diamond's clarity, it is viewed under 10x magnification by a trained gemologist. Small inclusions neither mar its beauty nor endanger its durability.

CUT

Many people confuse diamond cut with the shape of a diamond. Most diamonds are cut round with a full 58 facets, and a good cut, or make, has more scintillation, more sparkle. Although the cut of the diamond may affect the value of the diamond, the shape, is largely a matter of personal preference and does not affect the value significantly. It is the work of a master cutter that allows the diamond to be cut in such a way as to permit the maximum amount of light to be reflected through the diamond, and that's a great reflection on you. It is the cut that enables a diamond to make the best use of light.

CARAT

Carat weight is the weight of the diamond measured in carats. One carat is divided into 100 points, so a diamond of 75 points weighs 0.75 carats. Carat weight is the most obvious factor in determining the value of a diamond. But two diamonds of equal carat weights can have very different prices, depending on their quality. Diamonds of high quality can be found in all sizes.

GIA Diamond Grading Further Information