Fundamental particles are excitations of fundamental fields.
A quasiparticle is an excitation of a multi-particle system. The difference between these and elementary particles is that quasiparticles cannot exist outside of a multi-particle system.
A good example are semiconductor holes where many electrons that move in one direction result in the appearance of a positive hole moving in the opposite direction. This positive hole is termed a quasiparticle.
Other examples of quasiparticles are phonons, exitons and plasmons.
I’m not sure if there is a table$^1$ of quasiparticles. If there was, it would not be similar to the standard model since quasiparticles are a consequence of a system of particles, unlike the particles of the standard model.
$^1$ Edit: After answering another question regarding quasiparticles, I happened to come across this article on Wikipedia which lists and describes (with links to) the various known quasiparticles.