The original pion condensation was a liquid crystal like phase in neutron
matter conjectured to occur at high densities. The main neutron coupling
to pions is given by a term like $\vec \sigma \cdot \vec \nabla \Pi$,
where $\vec \sigma$ is the neutron spin operator and the $\Pi$ represents
the pion quantum field. This interaction gives an attraction when
the relative angular momentum of the nucleon and pion is 1, or a p-wave
attraction. If it results in a pion condensate it is sometimes called
a p-wave condensate. Similarly for the s-state with angular momentum 0.
The conjectured liquid crystal phase has a neutron liquid where the spin of the
neutrons tend to be aligned in layers, with opposite spin in opposite
layers as if there were an interaction $\vec \sigma \cdot [\hat z\cos(qz)]$.
That is the pion field with wave vector $q$ has a component that can
be viewed classically, or alternatively that field mode
has a macroscopic occupation number. This is then called a pion condensate.
If this drives a phase transition (rather than a cross over), then the
compressibility of the neutron matter will change dramatically
at the transition.
This change in state
will change things like the mass-radius relation of neutron stars.
LIGO observations on neutron star mergers etc. can give more
the equation, exotic states like these may be either observed or ruled out.