Particles have spin (S), angular momentum(J), charge, masses and other quantum numbers. Spin states don't mix with momentum components and so on. So we define particle as set of states which mix only among themselves under Poincare (translational and Lorentz) transformations.

A set of objects which transform under certain group of transformation is known as representation of that group. Actually the matrix representation of state and the operator itself both are considered as representation of the group.

Particle have some spin states, angular momentum states, momentum states, which mix only among themselves in presence of EM field and other external conditions. We want to have irreducible representations (because irreducible representation are building blocks to provide most general description of nature.) of the group so that there are no such states which transform only among themselves.

To have observables Poincare invariant (\(\bra{\psi}P^{\dagger }P\ket{\psi}= \bra{\psi}\ket{\psi}\)) implies that P shall be unitary. So we need unitary irreducible representations of the Poincare group.

So particle is set of states which transform under unitary irreducible representations of the Poincare group.
Similarly we can say that identical particles are the particles with same number of quantum numbers and which transform under same unitary irreducible representations of the Poincare group.

We have some finite dimensional representations of Poincare group which are scalar (tensor with rank zero),vectors (\(A^{\mu }\) tensor with rank 1), and higher rank tensors \(T^{\mu \nu }\) with 1, 4, 16, 64, components. But these representations are not unitary. By that we mean the Lorentz transformation matrix (\(\Lambda,e^{\frac{1}{2}\omega _{\mu \nu }M^{\mu \nu }}\)) is not unitary.

There are in fact no finite dimensional irreducible representations of the Poincare group. The unitary irreducible representations of the Poincare group were classified by Eugene Wigner , they are all infinite dimensional and are naturally described by fields.