Clarity refers the visual appearance of internal characteristics of a diamond that have naturally formed in the diamond.
Look into most diamonds with a jeweller's loupe (magnifying eyeglass) and you will see small "inclusions", also known as "nature's fingerprints". They look like small clouds or feathers but are usually invisible to the naked eye. Inclusions can affect the diamond's fire, but they also make your diamond unique and shouldn't always be seen as a fault.
The most sought after diamonds are those which are absolutely clear, free of any scratches, tiny characteristics or trace minerals that can detract from their purity and beauty. Those diamonds are the rarest and consequently the most expensive.
The best-value buy is what is known as the "eye-clean" diamond, typically the VS grades which do not contain visible inclusions.