Functor between categories with weak equivalance. A homotopical category is category with a distinguished class of morphism called weak equivalence. 

A class $W$ of morphisms in $\mathcal{C}$ is a weak equivalence if:
  
  
*
  
*All identities are in $W$. 
  
*for every $r,s,t$ which compositions
  $rs$ and $st$ exist and are in $W$, then so are $r,s,t,rst$
  

Given two homotopical categories $\mathcal{C}$ and $\mathcal{D}$, there is the usual notion of functor category $\mathcal{C}^{\mathcal{D}}$. But, there is also the category of homotopical functors from $\mathcal{C}$ to $\mathcal{D}$, which I am confused about. Note that, $F:\mathcal{C} \rightarrow \mathcal{D}$ is a homotopical functor if it preserve the weak equivalence. I am quoting from page 78 of a work by Dwyer, Hirschorn, Kan and Smith; called Homotopy Limit Functors on Model Categories and ... 

Homotopical functor category $\left(\mathcal{C}^{\mathcal{D}}\right)_W$ is the full subcategory of $\mathcal{C}^\mathcal{D}$ spanned by the homotopical functors. (What does it mean?)

Also, in the definition of homotopical equivalence of homotopical categories, we read that $f:\mathcal{C}\rightarrow \mathcal{D}$ and $g:\mathcal{D}\rightarrow \mathcal{C}$ are homotopical equivalence if their compositions $fg$ and $gf$ are naturally weakly equivalent, i.e. can be connected by a zigzag of natural weak equivalence. 
I do not know what the authors mean by a zigzag of natural equivalence. Please, help me with these concepts and tell me where I can learn more about them.
Thank you.

Maybe you're beginning your journey through model, localized, homotopy categories by a steep way. I would try this short paper first: W. G. Dwyer and J. Spalinski.
