What is a microstate, macrostate and thermodynamic probability in statistical mechanics?

A ‘microstate’ refers to a description of the system which relies on the states of each element of the system. Applied to a thermodynamic system, each microstate $M_i$ of the system is a set of positions $\{q_i\}$ and velocities $\{\dot q_i\}$ for $i = 1,\ldots,3N$ (in three dimensions, add another set of coordinates for internal degrees of freedom, such as rotation) which describe the position and velocity of each particle. As you can imagine, for large $N$ (say, $N = 10^{23}$), this gets out of hand. Furthermore, the probability that the system is in microstate $M_i$ is quite low as there are many, many different microstates the system could occupy.

A ‘macrostate’ on the other hand is a state description relying on the macroscopic properties of the system: it’s temperature, pressure, volume, internal energy and such. For each macrostate, there are many, many microstates which result in the same macrospace: for example, if you interchange velocity (but not position) of two gas particles, the macrostate does not change, but you have a different microstate.