Examples of faithfully flat modules

Formal properties
The tensor product of two faithfully flat modules is faithfully flat.
If $M$ is a faithfully flat module over the faithfully flat $A$-algebra $B$, then $M$ is faithfully flat over $A$ too.
An arbitrary direct sum of flat modules is faithfully flat as soon as at least one summand is.
(But the converse is false: see caveat below.)

Algebras
An $A$-algebra $B$ is faithfully flat if and only if it is flat and every prime ideal of $A $ is contracted from $B$, i.e. $\operatorname{Spec}(B) \to\operatorname{Spec}(A)$ is surjective.
If $A\to B$ is a local morphism between local rings, then $B$ is flat over $A$ iff it is faithfully flat over $A$.

Caveat fidelis flatificator
a) Projective modules are flat, but needn't be faithfully flat. For example $A=\mathbb Z/6=(2)\oplus (3)$ shows that the ideal $(2)\subset A$ is projective, but is not faithfully flat because $(2)\otimes_A \mathbb Z/2=0$.

b) A ring of fractions $S^{-1}A$ is always flat over $A$ and never faithfully flat [unless you only invert invertible elements, in which case $S^{-1}A=A$].

c) The $\mathbb Z$-module $\oplus_{{{\frak p}}\in \operatorname{Spec}(\mathbb Z)} \mathbb Z_{{\frak p}}$ is faithfully flat over $\mathbb Z$. All summands are flat, however none is faithfully flat.

If $A$ is a noetherian ring, $I\subseteq A$ is an ideal, and $\widehat{A}$ is the $I$-adic completion of $A$, if $I$ is contained in the Jacobson radical of $A$ (for example, if $A$ is local) then $A\to \widehat{A}$ is faithfully flat.

Let $f:A\to B$ be a flat ring homomorphism. If $q$ is a prime ideal of $B$, then $p=f^{-1}(q)$ is a prime ideal of $A$. Furthermore, we have an induced homomorphism of rings $\overline{f}:A_p\to B_q$ such that the composition $A\to B\to B_q$ (the first arrow is $f$ and the second arrow is the localization homomorphism $B\to B_q$) is equal to $A\to A_p\to B_q$ (the first arrow is the localization homomorphism $A\to A_p$ and the second arrow is $\overline{f})$. (Of course, this follows from the universal property of localization; every element of $A$ not in $p$ is mapped to an element of $B$ not in $q$.)

In this situation, $\overline{f}:A_p\to B_q$ is a faithfully flat ring homomorphism. (Proof: We use the transitivity of flatness. In particular, $B_q$ is a local ring of $B_p$ and is therefore flat over $B_p$ and $B_p$ is flat over $A_p$ since $B_p\cong B\otimes_{A} A_p$ and flatness is preserved under base change. Faithful flatness follows since $pA_p$, the unique maximal ideal of $A_p$, is mapped into $qB_q$, a proper (in fact, maximal) ideal of $B_q$, under $\overline{f}$.)

In particular, the induced map on spectra $\overline{f}^{*}:\text{Spec}(B_q)\to \text{Spec}(A_p)$ is surjective.

If $A$ is a coherent ring, any ring of power series $A[[X_1,\dots,X_n]]$ is faithfully flat over $A$.

Also, the direct sum of a flat $A$-module and a faithfully flat $A$-module is faithfully flat.

The direct sum $\bigoplus\limits_{\mathfrak m\in\operatorname{Max}A} A_{\mathfrak m}$ is faithfully flat over $A$.