What are some nontrivial examples of schemes whose Brauer group is trivial? I am aware that rational varieties have trivial Brauer group and retract rational varieties in the sense of Saltman also do. Can someone give me an example of a variety outside this class whose Brauer group is trivial?
There are plenty! A smooth complex projective variety with $H^2(X,\mathcal{O}_X)=0$ and $H^3(X,\mathbb{Z})$ torsion free has trivial Brauer group. For instance any complete intersection in $\mathbb{P}^n$ of dimension $\geq 3$ will have trivial Brauer group  and many others.
Every curve over an algebraically closed field has trivial Brauer group, by Tsen's Theorem.

$\begingroup$ And likewise for every (smooth proper) curve over a finite field by class field theory (though it seems that the OP is only asking about varieties over an algebraically closed field). $\endgroup$ – user76758 Feb 4 '14 at 16:47
This is not a variety, but an interesting example: Let $K$ be a number field with ring of integers $\mathfrak{O}_K$. Let $U$ be an open subscheme of $\mathrm{Spec}\,\mathfrak{O}_K$. Then one has an exact sequence $$0 \to \mathrm{Br}(U) \to \mathrm{Br}(K) \to \bigoplus_{v \in U}\mathrm{Br}(K_v).$$ Now use the AlbertBrauerHasseNoether theorem.