What is meant by "fairly" when used as a quantifier for the solubility of a chemical?
It seems that you understand the concept very well and that this is just a minor confusion about word fairly. 

If you would characterize the solubility of a compound (solid, liquid, or gas) by words, rather than by giving the exact solubility in $\pu{mg\cdot mL^{-1}}$, you would probably come up with terms like


highly soluble
soluble
slightly soluble
insoluble


So, fairly means that the solubility isn't excellent, but it's not that bad either. It is acceptable, which of course isn't very precise.


  Is it that fairly soluble in water means that a part of it dissolves while another part doesn't dissolve?


That one part does not dissolve will happen in all aqueous solutions when you add to much of the compound that you want to dissolve. Just think in sodium chloride, which has a solubility of about $\pu{390 g\cdot L^{-1}}$ in water. When your solution is saturated, it can't take any more. If you add another spoon of the salt, it will not dissolve and precipitate.