Cut - the only influence man has on a diamond, the rest is up to nature. The better the cut the better it handles light and sparkles more. A well cut diamond is one that is cut with the most ideal proportions. Unfortunately looking online you simply cannot tell the quality of the cut as you need to see the diamone in person.
Colour - many diamonds you find on the high street are actually tinged yellow, you perhaps wouldn't notice straight away by looking at it but next to a whiter diamond it's quite obvious. The colour grades start at D through to Z. For a diamond to be White and not have a hint of yellow it must be graded colour D,E,F,G or H. The closer to D the more expensive.
Clarity - or 'natures fingerprints'. These are small marks in a diamond that can range from black marks, 'clouds' lines or any imperfections. A certified 'flawless' diamOnd is the best and most expensive. However if you are not loaded then a perfectly acceptable to most would be no lower than Si2. Si2 and above may have flaws but they will not be able to be seen with the naked eye, you would need x 10 magnification.
Carat weight - the actual weight of the diamond. For example you could get a whole carat of diamond but if the other qualities are poor then it will be worth much less than a smaller diamond with higher qualities.
For me it's all about balance and getting a good combination of all of the above
Hth