  The Brauer algebra has a basis of diagrams and these generate a monoid $H$
consisting of scalar multiples of diagrams. Following a recent paper by
Kudryavtseva and Mazorchuk, we define and completely determine three types of
conjugation in $H$. We are thus able to define modular characters for Brauer
algebras which share many of the properties of Brauer characters defined for
finite groups over a field of prime characteristic. Furthermore, we reformulate
and extend the theory of ordinary characters for Brauer algebras as introduced
by Ram to the case when the Brauer algebra is not semisimple.
