As the other answers mentioned, the first crucial distinction you must make is that some properties refer to a topological dynamical system $(X,T)$, while others refer to a measure-preserving dynamical system $(X,T,\mu)$. Thus there are two different sets of definitions. Let me attempt a sketch at some of the relationships within each set.
First suppose you have a topological dynamical system $(X,T)$. Then the picture is the following. Let me add here the caveat that I've thrown this together somewhat quickly to illustrate the general shape of the relationships, so this is subject to the disclaimers below.
Counterexamples 1-9 illustrating the strict containments are as follows. (These may not be the simplest or the earliest counterexamples in each case, and I welcome corrections or improvements. This is based on some quick googling for things not already in my memory.)
1. $X = \Sigma_2 \times \{a,b\}$, the direct product of a full two-shift with a period-two orbit, where the dynamics is $\sigma\times S$, with $\sigma$ the shift map and $S$ the map interchanging $a$ and $b$.
2. $X=\Sigma_2$.
3. Constructed by Bassam Fayad, Topologically mixing and minimal but not ergodic, analytic transformation on $\mathbb{T}^5$, 2000.
4. Constructed by Furstenberg, Strict ergodicity and transformation of the torus, 1961.
5. ** Here I do not know an explicit example, but I expect one exists. Can anyone provide a reference? **
6. Karl Petersen, A topologically strongly mixing symbolic minimal set, 1970.
7. Rotation of the circle by an irrational angle.
8. ** Again I do not know an explicit example, but I expect one exists. Can anyone provide a reference? **
9. North-south map: a map $T\colon [0,1]\to [0,1]$ with fixed points at $0,1$ and such that $T(x) < x$ for all $x\in (0,1)$.
Then there are the ergodic properties: those that depend on a system preserving an measure $\mu$. For these one has the ergodic hierarchy.
Ergodic hierarchy http://www.math.uh.edu/%7Eclimenha/pics/ergodic-properties.png
It is very often the case that one wishes to study a topological dynamical system as a measure-preserving system by equipping it with an invariant measure, and in this case it is quite reasonable to ask about the relationships between the two different classes of properties. But this depends on which invariant measure you choose, because in general there may be very many of them. One may ask what properties of $(X,T)$ let you pick invariant measures $\mu$ with certain nice properties, and this is a whole different story which would expand this answer far beyond the bounds of propriety.