A theoretical definition of "elementary particle" is that it can be described within a quantum field theory based on an irreducible local representation of the Poincare group (the emphasis is on "irreducible", defining what we mean by "elementary").
As said, this is a theoretical definition. In practice it's not fully valid, because you can describe also composite particle, quite well in terms of what we defined to be an "elmentary" particle as long as you don't look at it too closely, i.e., as long as you don't scatter it with other particles at an energy high enough to resolve its structure it has becose of its compositeness. That's in fact a gift of nature, because it allows us to describe the "relevant" low-energy degrees of freedom in terms of elementary quantum fields, socalled effective field theories.
An important example is the standard model of elementary particles. Almost nothing concerning matter in everyday form can be described using Quantum Chromodynamics, which is the part of the standard model that describes the strong interactions among quarks and gluons. Never ever we have found a free quark or gluon, which are thought to be elementary at any energies so far available to be studied on Earth (including the largest energies available at the LHC at CERN). Instead, what we find in the very low energy regime, applicable to everyday life, are hadrons, i.e., (in fact pretty complicated not yet fully understood in all details) bound states of quarks and gluons, which are color neutral. So far we've seen quark-antiquark states (the mesons) and three-quark states (the baryons) bound together with a lot of other (virtual) quarks and gluons.
Fortunately, it's not necessary to describe the hadrons in all this details as long as you don't resolve this complicated bound-state structure. You can use effective models, where the hadrons are described by elementary quantum fields, using the (accidental) symmetries of QCD, most importantly the chiral symmetry in the light-quark sector.
Only if you use high-energy probes, you'll find out that the hadrons are composite objects. Historically the quarks where postulated by Gell-Mann and Zweig to understand the zoo of hadrons observed in the early 1960ies. At this time they didn't think that quarks with their -1/3 and +2/3 elementary charges exist at all but that you can use a mathematical scheme based on group theory to bring order in the zoo of hadrons. Somewhat later at SLAC by shooting high-energy electrons on protons one (Feynman) figured out that the cross section is compatible with the scattering on point-like particles making up the protons. The socalled parton model was born. Particularly it explained the socalled Bjorken scaling of cross sections.
The parton model had, however, a problem: One had to asssume that there exist hadrons which violate the Pauli principle. On one hand the quarks had to be clearly spin-1/2 particles which should obey the Pauli exclusion principle, on the other hand the parton model had to assume that there are bound states of these quarks that violate it (e.g. the ##\Omega## baryon consisting of three strange quarks).
The solution came with the idea that each quark comes in three "colors", which finally lead to the modern theory describing the strong interaction, quantum chromodynamics, which could also explain the deviations from Bjorken scaling. As a non-Abelian gauge theory it has the remarkable feature of "asymptotic freedom", i.e., the coupling constant becomes small if particles are colliding with high energy-momentum exchanges. This explains why one can consider the deep inelastic scattering of electrons observed at SLAC as scattering on quasi free partons within the hadrons. On the other hand it also explains why one cannot use perturbation theory of QCD at low collision energies: The coupling constant becomes large then, and perturbation theory becomes unreliable. That's why we have to use effective hadronic models to describe the interactions with hadrons at low energies or lattice QCD simulations which evaluate the QCD action on a discrete space-time grid without using perturbation theory.
