Just to give you a very 'quick and dirty' starting point, bearing in mind that I did my master's degree work some years ago and there are a few others here who could give better answers, at least if they want:
"Fermi fluid" and "Fermi gas" refer to states of matter where the dominant behavior of the matter can only be understood in terms of quantum mechanics (classically defined atoms do not exist and are either crushed by pressure, split up by heat, or both). Such substances are governed by what is called "Fermi-Dirac statistics." Basically, Fermi-Dirac statistics are set up on the proposition that no two things can occupy the exact same 'state' at the same time, where 'state' is construed to include both physical location AND other properties that particles have in quantum mechanics.* Two particles can be in the same place, but only if they have different amounts of energy stored up in their physical state, are spinning with different amounts of angular momentum, et cetera.
Electrons are a good example of a particle that obeys Fermi-Dirac statistics, which is why (for example) the orbital shells in an electron* can "fill up" with electrons. There are only so many possible combinations of angular momentum and 'spin' available to an electron with a specified amount of potential energy in orbit around an atom, so there are only so many 'slots' for the electron to occupy.
Fermi "fluid" and "gas" would have different properties governed by the same basic set of relevant laws. Just as in real life liquids and gases have different properties that are consequences of the same laws being applied to the same substance at different temperatures and pressures. The difference is that neutrons are very much not molecules in a gas and don't behave like molecules in a gas. They behave like fermions (because they are, they obey the "two neutrons can't have the same state at once" rule). But to know what fermions will do under certain conditions, you use Fermi-Dirac statistics.
The dominant force driving the behavior of matter under Fermi-Dirac statistics is that while all particles are free to "wash around" and go wherever they 'like' within the material... The physics requires that the lowest-energy 'states' fill up first, then higher and higher energy states are the only ones available. This governs how the material stores and exchanges energy with other substances, and what processes can occur within the material.
Just like how the behavior of ice, water, and steam is governed by temperature, how much energy is available for individual water molecules to bounce around, and to what extent the water molecules "stick together." In ice, the molecules have little energy individually and are firmly stuck together; in water they can move around each other freely and the material 'flows,' but cannot be squeezed or compressed; in gas the molecules move around with great speed and do not stick together at all.
*[By contrast there is also 'Bose-Einstein' statistics, which describes the behavior of particles that ARE allowed to occupy the same 'state' at the same time.
NOW, that leaves 'hyperons,' 'delta particles,' and so on.
Most of the exotic particles referred to are things that normally would undergo radioactive decay very quickly, so we only observe them in particle accelerators as byproducts of slamming protons together at stupidly high energies.
However, the reason these particles undergo decay is because doing so is 'favored' in terms of energy. An unattached neutron not part of a nucleus, moving through empty space, occupies a 'higher energy' state than it would if the same neutron split up, decaying into a proton (which carries away most of the mass) an electron (to balance out the electric charges), and an antineutrino and a gamma ray (to carry away the remaining energy, make sure momentum balances out, et cetera)
Free, unattached neutrons have an average lifetime of about ten minutes before this spontaneously happens to them.
But if we take the same neutron and stick it into an atomic nucleus, the decay process is no longer favorable. The neutron will not decay on its own, as long as the nucleus itself is stable and in balance.
So we can imagine that other subatomic particles which would decay even more quickly in their 'free' state might become stable when 'bound' under unusual conditions of high density, high pressure, high energy, et cetera.
This is where we get the prediction that things like hyperons and delta particles would exist in the core of a neutron star. Because the pressures and energies down there SHOULD be so great that these short lived, ultra-unstable particles would be able to take on a more lasting existence. Or be spawned into existence so frequently that they are constantly found there. Not sure which from the information available.