Clarity, as the name implies, refers to how clear the material is that the stone is made of, and what surface imperfections it may have. These have a direct effect on how light passes into a diamond and bounces around within it before finally exiting.
The internal flaws are called inclusions and they appear for several reasons, usually either foreign material within the diamond or even another diamond crystal inside it or an imperfection such as a tiny crack in the structure of the material itself.
Surface imperfections are popularly called blemishes and they can be either major or minor and will affect the value of the stone accordingly.
Clarity is determined by the number, size, color, location, orientation and visibility of the inclusions and blemishes as seen under a 10X magnification. As mentioned earlier, flawless stones are very rare, so it is likely that you will be considering stones that are less than perfect. That's perfectly fine, particularly as many of these imperfections are scarcely visible to the naked eye and often only when you are scrutinizing the diamond very closely.