Suppose that $(A, d_A),(B,d_B),(C,d_C)$ are (unbounded) differential graded algebras and that $f:A \to C$ and $g:B \to C$ are homomorphisms of differential graded algebras. What is (or how do we compute) the pullback:
$$ \begin{array}{cc} A \times_C B & \longrightarrow & B \\ \downarrow & & \downarrow\\ A & \longrightarrow & C \end{array} $$
in the category of differential graded algebras? Is it done differently with commutative diff. graded algebras?
Moreover is there a 'standard reference' for pullbacks or all kinds of 'common' (co)limits? Not very time economical if everyone computes them by herself.
Pullbacks (in fact, all limits) in categories of algebraic structures are created by the forgetful functor to $\mathsf{Set}$. Such basic facts can be found in Mac Lane's CWM, but you can also just prove it. The category of differential graded algebras is of course algebraic.
Thus, we have $(A \times_C B)_n = A_n \times_{C_n} B_n$, with componentwise operations, and componentwise differential, i.e. $d(a,b) = (da,db)$.
For differential graded algebras, I think there is also the notion of a homotopy pullback. This is more complicated, but also more natural for some applications in homotopical algebra.