For Poincaré algebra there are (as far as I know) two different approaches to find its representations. In the first approach one begins from a finite dimensional representation of (complexified) Lorentz algebra, and using it one constructs a representation on the space of some fields on Minkowski space. Representation so obtained is usually not irreducible and an irreducible representation is obtained from it through some differential equation. E.g. space of massive Dirac fields satisfying Dirac equation form an irreducible representation of Poincaré group (added later : last statement is not quite correct).
Another approach is to find (irreducible, unitary) Hilbert space representation of identity component of Poincaré algebra by so called "Little group method". This is what Weinberg is doing in pages 62-64 in volume 1 of his QFT book. Idea of this approach is following --
In momentum space fix a hyperboloid $S_m=\{p|p^2=m^2,p_0 \geq 0\}$ corresponding to a given (nonnegative) mass $m$. (note : here I am using signature $(1,-1,-1,-1)$)
Choose a 4-momentum $k$ on $S_m$. Let $G_k$ be the maximal subgroup of (the identity component) of the Lorentz group such that $G_k$ fixes $k$. i.e. for each Lorentz transformation $\Lambda\in G_k$ we have $\Lambda k=k$. $G_k$ is called little group corresponding to 4-momentum $k$.
Let $V_k$ be a fixed finite dimensional irreducible representation of $G_k$ (or double cover of $G_k$)$^{**}$. Fix a basis of this vector space $|k,1\rangle,|k,2\rangle,\ldots,|k,n\rangle$ where $n$ is (complex) dimension of $V_k$ {note that $k$ is a fixed vector, and not a variable.}
Now for every other $p\in S_m$ introduce a vector space $V_p$ which is spanned by the basis $|p,1\rangle,|p,2\rangle,\ldots,|p,n\rangle\;.$
Hilbert space representation of (the identity component of) the Poincaré group is now constructed by gluing these vector spaces $V_p$'s together. This is done as follows :-
i) Define $H$ to be direct sum of $V_p$'s.
ii) For every $p\in S_m$ fix a Lorentz transformation $L_p$ that takes you from $k$ to $p$, i.e. $L_p(k)=p$. Also fix a number $N(p)$ (this is used for fixing suitable normalization for the basis states). In particular, take $L_k=I$.
iii) Define operator $U(L_p)$ corresponding to $L_p$ on $V_k$ as :-
$U(L_p)|k,\sigma\rangle =N(p)^{-1}|p,\sigma\rangle,\:\sigma=1,\ldots,n\tag1$
This only defines action of $L_p$'s on subspace $V_k$ of $H$. But in fact this definition uniquely extends to the action of whole of (identity component of) Poincaré group on the whole of $H$ as follows --
Suppose $\Lambda$ be ANY Lorentz transformation in the identity component of the Lorentz group, and $|p,\sigma\rangle$ be any basis state. Then (all the following steps are from Weinberg's book):
\begin{align}U(\Lambda)|p,\sigma\rangle &= N(p) U(\Lambda) U(L_p)|k,\sigma\rangle\,\,\,\,\,\, \textrm{using def. (1)}\\ &= N(p) U(\Lambda.L_p)|k,\sigma\rangle \,\,\,\, \textrm{(from requiring}\,\, U(\Lambda) U(L_p)=U(\Lambda.L_p))\\ &= N(p) U(L_{\Lambda p}.L_{\Lambda p}^{-1}.\Lambda.L_p)|k,\sigma\rangle\\ &= N(p) U(L_{\Lambda p})U(L_{\Lambda p}^{-1}.\Lambda.L_p)|k,\sigma\rangle\;.\end{align}
Now note that $L_{\Lambda p}^{-1}.\Lambda.L_p $ is an element of $G_k$ {check it} and $V_k$ is irreducible representation of $G_k$. So $U(L_{\Lambda p}^{-1}.\Lambda.L_p)|k,\sigma\rangle$ is again in $V_k$; and from (1) we know how $U(L_{\Lambda p})$ acts on $V_k$; thus we know what is $U(\Lambda)|p,\sigma\rangle\;.$
Summarizing, the idea of little group method is to construct irreducible Hilbert space representations of the identity component of Poincare group starting from finite dimensional irreducible representations of the Little group corresponding to a fixed four momenta.
$^{**}$ If $V_k$ is not a proper representation of $G_k$ but is a representation of the double cover $\mathcal{G}_k$ of $G_k$ then we'll also need to specify a section $G_k\to \mathcal{G}_k$ of the covering map so that we know how $G_k$ acts on $V_k$.
As you probably know, the Lie group of physical transformations of a quantum system acts on the Hilbert space of states of the system by means of a (strongly-continuous projective-) unitary representation of the group. $G \ni g \mapsto U_g$. This action is effective also on the observables of the system, represented by self-adjoint operators: The action of $g$ on the observables $A$ is $U_gAU^*_g$. The latter represents the observable $A$ after the action of the transformation $g$ on the physical system. This transformation has a twice intepretation. We can imagine that either it acts on the system or on the reference frame, our choice does not matter in this discussion.
Now let us focus on physics. There are natural elementary systems, called elementary particles. These systems are completely determined by fixing some real numbers corresponding to the values of some observables. Within the most elementary version of the story, these numbers are the mass $m$ which
may attain a few positive numbers experimentally observed ad recorded,
and the spin $s$ which may attain any number in $\{1/2, 1, 3/2,...\}$. Different values of the pair $(m,s)$ mean different particles.
These numbers have the property that they are invariant under the action of the most general symmetry group, I mean the (proper orthochronous) Poincaré group. A type of particle has the same fixed numbers $m$ and $s$ independently from the reference frame we use to describe it and the various reference frames are connected by the transformation of Poincaré group.
Passing to the theoretical quantum description of an elementary particle, in view of my initial remark, we are committed to suppose that its Hilbert space supports a representation of Poincaré group ${\cal P} \ni g \mapsto U_g$ (I omit technical details). Moreover there must be observables representing the mass $M$ and the spin $S$ that, on the one hand they must be invariant under the action of the group, i.e., $U_gM U_g^* =M$ and $U_gS U_g^* =S$ for every $g \in \cal P$. On the other hand they must assume fixed values $M=mI$ and $S=sI$.
Wigner noticed that a sufficient condition to assure the validity of these constraints is that ${\cal P} \ni g \mapsto U_g$ is irreducible.
Indeed, $M$ and $S$ can be defined using the self-adjoint generators of the representation, since they are elements of the universal enveloping algebra of the representation of the Lie algebra of $\cal P$ induced by the one of $\cal P$ itself. As expected, one finds $U_gM U_g^* =M$ and $U_gS U_g^* =S$ for every $g \in \cal P$. But, if $U$ is also irreducible, re-writing the identities above as
$U_gM =M U_g$ and $U_gS =S U_g$ for every $g \in \cal P$, Schur's lemma entails that $M=mI$ and $S=sI$ for some real numbers $s,m$.
To corroborate Wigner's idea it turns out that the two constants $m$ and $s$ are really sufficient to bijectively classify all possible strongly-continuous unitary irreducible representations of $\cal P$ with "positive energy" (the only relevant in physics).
The mathematical theory of representations of ${\cal P}$ autonomously fixes the possible values of $s$ and they just coincide with the observed ones. The values of $m$ are not fixed by the theory of representations where any value $m\geq 0$ would be possible in principle, though not all $m \geq 0$ correspond to the masses of observed elementary particles.
If you have many elementary particles, the Hilbert space of the system is the tensor product of the Hilbert spaces of the elementary particles and there is a corresponding unitary representation of Poincaré group given by the tensor product of the single irreducible representations. Obviously, the overall representation is not irreducible.
ADDENDUM. I would like to specify that the irreducible representations of the group of Poincaré I discussed above are the faithful ones whose squared mass is non-negative. Moreover, there is another parameter which classifies the irreducible representations of Poincaré group. It is a sign corresponding to the sign of energy. Finally not all particles fit into Wigner's picture.
Best Answer
This is answered in depth in Weinberg's book on quantum field theory (Vol. I, Chapter 2).
Relativistic invariance means translation invariance and Lorentz invariance, hence - obviously - Poincare invariance, so that one has a representation of the Poincare group. Because of relativistic invariance and unitarity, the Hilbert space of a QFT carries a unitary representation of the Poincare group, and it splits (as any unitary representation) into a direct sum of irreducible ones. Being irreducible means being not further divisible, hence elementary. One can classify them, and finds that they describe single relativistic particles, hence elementary particles.
Irreducible representations have constant Casimirs, but the values of the constants do not always characterize the irrep. In particular, all massless irreducible representations of the Poincare group have the same values for the Casimirs but may differ in helicity.