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!):
FL: Completely flawless
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.
There are many different types of flaws. The best way to become acquainted with them is to look at lots of diamonds. The more common ones are as follows:
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.
One pitfall to avoid when shopping for clarity is the "clarity-enhanced" diamond. This is an artificial process used to "fix" the flaws on an otherwise good stone. Although a clarity-enhanced diamond can look nearly flawless (in some cases, it is impossible to detect the enhancement), it is intrinsically worth as much as a flawed stone. Furthermore, the durability is nowhere that of pure diamond. Be sure to confirm with your jeweler that the stone you are considering is not clarity-enhanced, and be sure to double-check this fact on the G.I.A. certificate. If you do not receive a G.I.A. certificate with your stone, ask the jeweler to state that the diamond is not clarity-enhanced on the appraisal.