What's the difference between an ergodic measure, ergodic sequence or ergodic mapping?

The notions of ergodic measure and ergodic mapping are more or less interchangeable. If you already have a probability space $(Omega, mathcal{F}, mathbb{P}),$ you would say that a measure-preserving transformation $T: X rightarrow X$ is ergodic if $mathbb{P}(A) in {0, 1}$ for all $A in mathcal{F}$ for which $T^{-1}(A) = A$ (this is one of many equivalent definitions). You could also say in this scenario that $mathbb{P}$ is ergodic (with respect to $T$). However, there are situations in which you are only given a mapping $T$ and are asked if there exists an ergodic measure with respect to $T$ (a very well known theorem shows that if $X$ is a complete metric space and $T$ is continuous, then indeed there exists an ergodic measure). So it really depends on the context, really. As for ergodic sequences, I believe that the wikipedia page on ergodic sequences is a good reference. I hope this helps. :)