$ \color{red}{P(1, 3)} = \color{blue}{T(4)} \color{magenta}{\rtimes} \color{green}{SO(1, 3)}$
The Poincare group is the mathematical tool that we use to describe the symmetry of special relativity.
The starting point for Einstein on his road towards what is now called special relativity was the experimental observation that the speed of light has the same value in all inertial frames of reference. This curious fact of nature was discovered by the famous Michelson-Morley experiment.
A symmetry is a transformation that we can perform without changing something. Therefore, the invariance of the speed of light under arbitrary changes of the frame of reference is a symmetry and we call this symmetry the Poincare group. The Poincare group contains all transformations that we can perform without changing the speed of light.
The $\color{red}{\text{Poincare group}}$ consists of $\color{blue}{\text{translations}}$ $\color{magenta}{\text{plus}}$ $\color{green}{\text{rotations and boosts}}$.
The Poincare Algebra
\begin{eqnarray} {}[J_i,J_j]&=&i \epsilon_{ijk}J_k, \nonumber \\ {} [J_i,K_j]&=& i\epsilon_{ijk} K_k, \nonumber \\ {} [K_i,K_j]&=& -i \epsilon_{ijk}J_k ,\nonumber \\ {} [J_i,P_0]&=&0, \nonumber \\ {} [J_i,P_j]&=& i\epsilon_{ijk} P_k, \nonumber \\ {} [P_0,P_i]&=&0, \nonumber \\ {} [K_i,P_0]&=&i P_i, \nonumber \\ {}[K_i,P_j]&=& i\ P_0 \delta_{ij} , \label{c} \end{eqnarray}
aspects.
The double cover of the Poincare group is the fundamental spacetime symmetry of modern physics and is a crucial component of the standard model of particle physics.
The Poincare group is the set of all transformations that leave the speed of light invariant. Thus, the Poincare group yields all possible transformations between allowed frames of reference. This is incredibly useful, when we want to write down fundamental laws of nature. The fundamental laws should be valid in all allowed frames of reference, otherwise they would be quite useless.
In practice, we can use our knowledge of all transformations inside the Poincare group to write down equations that are invariant under all these transformations. These equations then hold in all allowed frames of reference. This is such a strong restriction on the possible equations that is almost enough to derive the most important equations of fundamental physics: the Dirac equation, the Klein-Gordon equation and the Maxwell-Equations.
"The Hilbert space of one-particle states is always an irreducible representation space of the Poincare group. […] The construction of the unitary irreducible representations of the Poincare group is probably the most successful part of special relativity (in particle physics, not in gravitation theory, for which it is a disaster). It permits us to classify all kinds of particles and implies the main conservation laws (energy-momentum and angular momentum). […] The translation generators are responsible for the energy-momentum conservation laws, the rotation generators of the conservation of angular momentum, and the boost generators of the conservation of initial position. " from Reflections on the Evolution of Physical Theories by Henri Bacry
"The enlargement of the Lorentz group to the Poincare group was proposed [ 13] as a way of describing the quantum states of relativistic particles without using the wave equations. The states of a free particle are then given by the unitary irreducible representations of the Poincare group." from Deformed Poincare containing the exact Lorentz algebra by Alexandros A. Kehagias et. al.