Not uniquely ergodic transformation Could you teach me an example of NOT uniquely ergodic but ergodic transformation?
And when any continuous, measurable, and ergodic transformation on a topological space X is uniquely ergodic, how topological properties X has?
 A: For a transformation to have a unique ergodic measure is not that common as you seem to think. In fact for many examples (I would say for the vast majority) many ergodic measures exist. This is due to the fact the often there are periodic orbits giving rise to ergodic measures. Here are some classes of examples:
Rational rotations of the circle: There every (finite) orbit can be used as the support of an ergodic measure (the uniform distribution on this orbit).
Full shifts: Again there are many periodic orbits, so put the uniform measure on one of them and you get an ergodic measure. However there are many other ergodic measures for such a system (Bernoulli measures, Markov measures, Gibbs measures etc.).
Similarly endomorphisms of the circle, hyperbolic toral automorphisms, Markov shifts, sofic shifts etc. etc. etc. all have multiple ergodic measures.
On the contrary, the standard classes of uniquely ergodic systems cyclic permutations on a finite space, irrational rotations on the circle (a uncountable connected space) or more generally on a compact (abelian) topological group, sturmian subshifts or subshifts coming from primitive substitutions (both have totally disconnected phase spaces) etc.
So taking those uniquely ergodic examples, your second questions does not seem to have a good answer with respect to topology. Unique ergodicity has to do less with the topological properties of the phase space and more with the "rigidity" of the transformation.
There is one condition however: Think about what it would mean for a space to only allow for uniquely ergodic transformations. As I explained above more than one finite orbit is bad. So what are the spaces where no continuous transformation can have multiple finite orbits? Think of the most basic transformation, the one definable on every phase space... 
