I came across the term on several occasion while reading solid state physics and statistical mechanics. Didn't understand quite well. It'll be help if it's explained.
When you have a great number of interacting particles (such as electrons in a solid), it becomes impossible to try to describe each electron individually. We introduce the concept of quasiparticle to describe whan can be the low-lying excitations of the ensemble of particles.
For instance, If you take a metal at 0K, the ground state of the system is the Fermi sea. When you excite this metal with light, for instance, you can create a hole in the Fermi sea by making an electron jump to a state of higher energy. But due to the interactions with other electrons, this electron will eventually go back to its original state by recombining with the hole, and emitting a photon. We call this electron a "quasi-electron" because it has a finite lifetime.
You can think of quasiparticles as particles which are affected by the interactions with all the other particles surrounding it. I told you about the quasielectrons (which are fermions), but quasiparticles can be many other things : bosonic excitations such as magnons, plasmons, polarons... Are quasiparticles as well, in the sense that they are low-lying excitations of the ground state of a solid with a certain (long enough) lifetime.
Think about the wave-particle mechanism in quantum mechanics. Everytime you have a wave travelling in a medium, you can think of it as a particle with a given propagation law (dispersion law). In the case of phonons, you can show that they propagate with the sound speed on the given medium. So, you can think of it as a particle (a set of variable values travelling in space all together), just as like you think as something travelling through the stands on a stadium when you see a "mexican wave".