According to http://en.wikipedia.org/wiki/List_of_particles, the list of elemetary particles includes more than 30 particles (bottom of the page). Does the Standard Model explain (if yes, then how) how many of elementary particles we should expect?
In one of these video lectures on the Standard Model, Leonard Susskind explains what differentiates an elementary particle from a composite one. The actual difference is experimental and can a posteriori be rationalized with some theory.
Essentially, it comes to assessing similarities and differences between different subatomic particles. In particular, while protons and neutrons were thought to be elementary particles interacting via exchange of pions, later examination in the high energy realm made people realize that there were much heavier particles than the protons and neutrons but, apart from that, they would have essentially the same features. In addition, as the mass of these particles would increase, their corresponding angular momentum would grow as well following a power low known as Regge trajectory. From what people had tried, it seemed that an elementary particle description was incompatible with these observations and this gave rise to the idea that baryons are composite particles.
In general, a particle is said to be composite if it can exist at rest in an excited state (which is most of the time related to angular momentum like in atoms for instance).
Hence an elementary particle is a particle that cannot in principle exist in an excited state or for which the energy required to increase its angular momentum is at the Planck energy scale; electrons are one of those for instance.
As far as I know, the Standard Model aims at enumerating in a quite exhaustive way the different types of elementary particles and the way they interact with each other. It however doesn't prescribe how many particles there should be per type. If I am wrong please correct me.