I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be decomposed into smaller subsystems (the only invariant pieces are really big or really small), but once you know that a transformation is ergodic, why do you care if there is only one measure which it's ergodic with respect to or not?
Unique ergodicity is defined for topological dynamical systems and it tells you that the time average of any function converges pointwise to a constant (see Walters: Introduction to Ergodic Theory, th 6.19). This property is often useful.
Any ergodic measure preserving system is isomorphic to a uniquely ergodic (minimal) topological system (see http://projecteuclid.org/euclid.bsmsp/1200514225).