Let $X,Y Z$ be simplicial sets with two maps $a: X \to Z, b: Y \to Z$ and I would like discuss how the $n$-simplices of the new simplicial set $X \times_Z Y$. I found only a construction for direct products), but I'm not sure how it should look like for $X \times_Z Y$.
Can the $n$ simplices of $X \times_Z Y$ be constructed in pure set theoretic terms as $X_n \times_{Z_n} Y_n$ where $X_n, Z_n, Y_n$ are the sets of $n$-simplices of $X, Y, Z$?
If we recall what the $n$-simplices of $X$ are defined as the set of natural transformations $X_n := Nat(\Delta^n, X)$ where $\Delta^n$ is the standard $n$-simplex, a simplicial set defined as the functor $Hom_{\Delta}(-, [n]): \Delta \to $(Set), $[m] \mapsto Hom_{\Delta}([m], [n])$ where $[n]$ denotes the ordered set $\{0, 1, ... ,n \}$ of the first $(n + 1)$ nonnegative integers.
Therefore $(X \times_Z Y)_n= Nat(\Delta^n, X \times_Z Y)$. In general if $T, A, B, C \in $ (Set) and $a: A \to C, b: B \to C$, then
$$Hom_{(Set)}(T, A \times_C B)= \{ (f,g) \in Hom_{(Set)}(T,A) \times Hom_{(Set)}(T,B) \ \vert \ a \circ f= b \circ g \} $$
Can make similar identifications for $Nat(\Delta^n, X \times_Z Y)$ when $X, Y, Z, \Delta^n$ are not sets, but simplicial sets and Homs are replaced by natural transformation?
Recall that the category of simplicial sets is simply the presheaf category $Sets^{\Delta^{op}}$ where $\Delta$ is the simplex category.
In any presheaf category, limits and colimits are taken pointwise. That is, given a diagram $J \to Sets^{C^{op}}$, the limit of $J$ is given by $(\lim\limits_{j \in J} J(j))(x) = \lim\limits_{j \in J} J(j)(x)$ (with the obvious action on morphisms).
So yes, pullbacks (fibre products) of simplicial sets exist and are taken pointwise. That is, the $n$-simplices of a pullback are precisely the pullback of the $n$-simplices.