A microstate is a state of the system that can be uniquely specified by the state variables of the fundamental mechanics (quantum or classical mechanics at present).
If the mechanics is classical, the state space is spanned by the so-called phase variables: if there are N point particles in 3-space their position coordinates and the momentum coordinates are the basic variables (which span the phase space).
If the mechanics is quantum, then the state space of the systems is a (generalized) Hilbert space spanned by all the eigenvectors of the system Hamiltonian. A ket and its complex multiple denote the same physical state, the microstate of a quantum mechanical system is a one-dimensional subspace of a complex vector space.
Knowing free energy alone is not enough
Although that the beginning of the chapter gives an impression that the crucial purpose of statistical mechanics is to compute (the Helmholtz) free energy, as we will see and it is well recognized in thermodynamics that free energy cannot always completely specify a thermodynamic state of a macroscopic object. The complete description of a thermodynamic state is always in terms of thermodynamic coordinates (= internal energy + work coordinates see [T5]); thus all the variables are extensive variables. Temperature T cannot be used.
This also tells us that the most fundamental statistical mechanics is to compute S in terms of thermodynamic coordinates. Thus, the microcanonical ensemble is the privileged most fundamental ensemble.
The reader may wonder what the relation of this statement and the ensemble equivalence. The latter tells us that thermodynamic potentials with their natural variables can be obtained from each other by Legendre transformation. Thus, entropy as a function of thermodynamic coordinates can be reconstructed from the Helmholtz free energy with a function of T and work coordinates. However, there is no one-to-one correspondence between the states specified by the natural variables. For example, suppose a thermodynamic state by E and work coordinates. This state has a definite temperature T, so we could say T and the same set of work coordinates may be used to describe the state. However, there can be other thermodynamic state that can be specified by a distinct E (and the same work coordinates) but have the same T.
