Quasiparticles (related to fermions) and collective excitations (related to bosons) are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For example, as an electron travels through a semiconductor, its motion is disturbed in a complex way by its interactions with other electrons and with atomic nuclei. The electron behaves as though it has a different effective mass travelling unperturbed in vacuum. Such an electron is called an electron quasiparticle.
In another example, the aggregate motion of electrons in the valence band of a semiconductor or a hole band in a metal behave as though the material instead contained positively charged quasiparticles called electron holes. Other quasiparticles or collective excitations include the phonon (a particle derived from the vibrations of atoms in a solid), the plasmons (a particle derived from plasma oscillation), and many others.
The principal motivation for quasiparticles is that it is almost impossible to directly describe every particle in a macroscopic system. For example, a barely-visible (0.1mm) grain of sand contains around 1017 nuclei and 1018 electrons. Each of these attracts or repels every other by Coulomb’s law. In principle, the Schrödinger equation predicts exactly how this system will behave.
But the Schrödinger equation in this case is a partial differential equation (PDE) on a 3×1018-dimensional vector space—one dimension for each coordinate (x,y,z) of each particle. Directly and straightforwardly trying to solve such a PDE is impossible in practice. Solving a PDE on a 2-dimensional space is typically much harder than solving a PDE on a 1-dimensional space (whether analytically or numerically); solving a PDE on a 3-dimensional space is significantly harder still; and thus solving a PDE on a 3×1018-dimensional space is quite impossible by straightforward methods.
One simplifying factor is that the system as a whole, like any quantum system, has a ground state and various excited states with higher and higher energy above the ground state. In many contexts, only the “low-lying” excited states, with energy reasonably close to the ground state, are relevant. This occurs because of the Boltzmann distribution, which implies that very-high-energy thermal fluctuations are unlikely to occur at any given temperature.
Quasiparticles and collective excitations are a type of low-lying excited state. For example, a crystal at absolute zero is in the ground state, but if one phonon is added to the crystal (in other words, if the crystal is made to vibrate slightly at a particular frequency) then the crystal is now in a low-lying excited state. The single phonon is called an elementary excitation. More generally, low-lying excited states may contain any number of elementary excitations (for example, many phonons, along with other quasiparticles and collective excitations).
By investigating the properties of individual quasiparticles, it is possible to obtain a great deal of information about low-energy systems, including the flow properties and heat capacity.
In the heat capacity example, a crystal can store energy by forming phonons, and/or forming excitons, and/or forming plasmons, etc. Each of these is a separate contribution to the overall heat capacity.