What is an example of a second-order markov chain?

There is nothing radically different about second order Markov chains: if $P(x_i|x_{i-1},..,x_1)=P(x_i|x_{i-1},..,x_{i-n})$ is a "n-th order Markov chain", we can still interpret it as a first order Markov chain, on the space of combinations of $n$ states, i.e. $S^n$, if $S$ is the set of values $x_i$ takes: just write $P(x_i|x_{i-1},..,x_1)=P(x_i|x_{i-1},..,x_{i-n})=P((x_i,x_{i-1},..,x_{i-n+1})|(x_{i-1},x_{i-2},..,x_{i-n}))$. Thus a second-order markov chain is just one which takes into account the two previous states.

The markov property specifies that the probability of a state depends only on the probability of the previous state.

You can "build more memory" into the states by using a higher order Markov model.

In an $n$th order Markov model

$$P(x_i | x_{i-1}, x_{i-2},..., x_1) = P(x_i | x_{i-1},..., x_{i-n} ) $$

Example of a second order MC