Let $\Sigma=\{s_1,\dots,s_m\}$ be a finite list of symbols, and put $X=\Sigma^\mathbb{Z}$.
Consider the left two-sided shift $T:X\to X$ given by $T(x_n)=(x_{n+1})$. Given an $m$-dimensional vector $\vec{p}=(p_1,\dots,p_m)$, we can construct a measure on $\Sigma$ by $\sum_{i=1}^m p_i \delta_{s_i}$, which then generates an infinite product measure on $X$. Such a measure is called a Bernoulli shift, and is ergodic for $T$.
My question is: given a periodic orbit for $T$, that is, an element $x\in X$ with $T^k x=x$ for some $k$, can we construct a measure supported on the orbit $\{T^i x:0\le i<k\}$ which is also ergodic for $T$? Further, are there $T$-ergodic measure which are not of either kind? I'm not sure on how to start looking, a hint would be very welcome!
There are many other ergodic measures. The first "natural" measures after Bernoulli measures are Markov measures, see for example https://en.wikipedia.org/wiki/Subshift_of_finite_type.
After that you can consider more intricate rules such as $2$-step Markov measures, etc. This would mean that you give probabilities to pairs and than transition probabilities from a pair to a second pair (not intersecting the first).
On the other hand, in view of general results of ergodic theory, if you consider any closed invariant subset of $\Sigma$ you know that there are ergodic full entropy measures for the restriction of the shift to that set. Not however that the support (as in the case of Markov measures) is smaller than $\Sigma$.
Your measures supported on periodic orbits are on the other extreme since the restriction of the dynamics to them has zero entropy.
Other than this you can consider equilibrium measures of any Hölder continuous function, which gives many more examples (and there are uncountably many of these functions that are not cohomologous). Unfortunately these measures are not easy to write down explicitly, it really depends on what you are looking for.