Ergodic theory is, put simply, the statistical and qualitative behavior of measurable group and semigroup actions on measure spaces.

The group meant here is usually N (the set of natural numbers, used a great deal in number theory); R, which denotes the Real numbers, such as , 2.3, 1/7, e; R+, which is obviously the collection of positive R; or Z (the ring of integers, i.e. …, -2, -1, 0, 1, 2, …).

Originating in the work of Boltzmann in statistical mechanics problems where time- and space-distribution averages are equal, ergodic theory has developed considerably since. It’s wonderfully obtuse theory, used for things such as:-
“keeping one's feet dry ("in most cases," "stormy weather excepted") when walking along a shoreline without having to constantly turn one's head to anticipate incoming waves.
In mathematics, ergodic theory turned up first in the work of, amongst others, the great von Neumann, Birkhoff and Koopman in the 1930s.
In the last seventy years or so it has grown to vast dimensions and has applications as varied as to statistical mechanics, number theory, differential geometry and functional analysis.
In addition, ergodic theory is also applied to itself in some cases, which is a lovely Godelian sort of thing to happen in pure mathematics.

It's not quite the same as Dynamical System theory (although it's worth comparing), which essentially measures how a system changes over time; bluntly, ergodic theory measures more how a system functions in a space.

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