Clarity is a measure of the number and extent of the flaws in the diamond. Generally speaking, the fewer the flaws, the more valuable the diamond. Completely flawless diamonds are extremely rare -- only a few hundred "FL" diamonds are produced per year worldwide.

There are several grading systems used to describe clarity. By far, the most popular is the Gemological Institute of America's (G.I.A.) scale, which ranks diamonds as Flawless (FL), Internally Flawless (IF), very very slightly included (VVS), very slightly included (VS), slightly imperfect (SI), and imperfect (I):

Although seemingly subjective, the G.I.A. scale has specific criteria that are used to differentiate between the different grades (what's the difference between "very very" slight and "very" slight anyway!):

IF: Internally flawless; only external flaws are present, which can be removed by further polishing the stone

VVS1 - VVS2: Only an expert can detect flaws with a 10X microscope. By definition, if an expert can see a flaw from the top of the diamond, it is a VVS2. Otherwise, if an expert can only detect flaws when viewing the bottom of the stone, then it is a VVS1

VS1 - VS2: You can see flaws with a 10X microscope, but it takes a long time (more than about 10 seconds).

SI1 - SI2: You can see flaws with a 10X microscope

I1 - I3: You can see flaws with the naked eye. Consider avoiding I2-I3 diamonds.

Pinpoint: A very small white dot on the surface of the stone. By far, the most common flaw.

Carbons: A very small black dot on the surface of the stone. Less common than pinpoints.

Feathers: Small cracks within the stone, similar in look to broken glass. Small internal feathers are harmless (other than lowering the clarity rating of the diamond), but large feathers can become a problem because the crack can grow as the diamond ages.

Clouds: Hazy areas within the diamond, actually made up of many small crystals that are impossible to see individually.

Crystal Growth: A small crystalline growth within the diamond. Looks like a small diamond within the big diamond.

Unfortunately, clarity is very difficult to judge accurately by an inexperienced consumer, so your best bet is to gain an education first by looking at lots of diamonds before making a purchase. Any good jeweler will spend the time you need to get comfortable judging the clarity of your stone -- ask different jewelers to point out the flaws in several stones until you can detect pinpoints and other flaws by yourself.

Many people make clarity the least "important" of the 4 Cs when purchasing their diamonds. The rationale is obvious -- when your partner shows the ring to all her friends, the likelihood that one of them will pull out a 10X microscope to examine the flaws on her diamond are very slim. Given that, why spend a lot of money on a VVS1 diamond when an SI2 will look exactly the same to the naked eye?
If you're purchasing an emerald cut (or any other step cut), consider purchasing a diamond with clarity greater than SI1. Clarity flaws are much more readily visible in step cuts than in brilliant cuts.