What are predicates of a sentence?
Roughly, a sentence is just logical syntax: it's simply a string in the language of logic. E.g. a sentence $\phi$ could be
$$
    {\sf eats}({\sf cat}, {\sf head}({\sf tuna}))
$$
Above $\sf eats$ is a predicate symbol, $\sf head$ is a function symbol (which your book calls "functor"), while $\sf cat, tuna$ are constant symbols.

A structure defines how to interpret the symbols. For instance, we can choose the domain $D$ to be the set of real numbers $\mathbb{R}$. Then, we interpret the symbols as follows:
$$
\mbox{World $w$: }\qquad\qquad
\begin{array}{l}
  [\![ {\sf cat} ]\!]^c = 5 \\
  [\![ {\sf tuna} ]\!]^c = 4 \\
  [\![ {\sf squirrel} ]\!]^c = 2 \\
  \ldots \\
  [\![ {\sf head} ]\!]^f = g \mbox{ where } g(x) = \pi \cdot x \\
  \ldots \\
  [\![ {\sf eats} ]\!]^p = \{ \langle x , y \rangle \ |\ x &lt; y \}
\end{array}
$$
(A structure can define more symbols than those used in the formula, hence
the $\sf squirrel$ above.)

Constant symbols are interpreted as elements of $D$ by the interpretation $[\![-]\!]^c$.
Function symbols are interpreted as functions $D\rightarrow D$ by the interpretation $[\![-]\!]^f$.
Predicate symbols are interpreted as subsets of $D^k$ ($k$-ary relations, where $k$ is the number of arguments of the predicate) by the interpretation $[\![-]\!]^p$.

The sentence $\phi$ is satisfied by the structure $w$. This is because, if we interpret everything, we have
$$
  [\![ {\sf eats} ]\!]^p ([\![ {\sf cat} ]\!]^c, [\![ {\sf head} ]\!]^f([\![ {\sf tuna} ]\!]^c))
$$
which is
$$
  &lt; (5, g(4))
$$
i.e.
$$
  5 &lt; \pi \cdot 4
$$
which is true. The sentence $\phi$ is said to be satisfiable: there is some structure which makes it true.