Entanglement refers to one particle needing no time to react to change of another particle, not even the time for speed of light to travel. (that even gravitational force requires speed of light to travel so that two particles attract).
Is it by experiment that entanglement was found, or is it by some other theory that we can "deduce" entanglement, or both?
It is called quantum mechanics with its postulates and its equations of state. Quantum mechanics is a complete mathematical model of the micro world of particles atoms molecules and nano states.And it has been discovered to be the theory for all experimental data up to now, i.e. it has not been falsified.
A quantum mechanical solution of a system of particles by definition means that all the particles are "entangled", i.e. not independent of each other not only in kinematics ( as is the case with classical mechanics) but also with quantum numbers. It means that once there is a wavefunction $Ψ$ solution for the system each "event" building up the probability distribution for the system which is given by $Ψ^*Ψ$ will have the particles under consideration in specific quantum states.It is these quantum states that are used in the populist examples of "entanglement" .
Since mainstream physics states and can prove that all classical physics emerges from the quantum underlying framework,in principle everything is "entangled". BUT the complexity is such that it can be shown that macroscopic systems lose their coherence. This is mathematically modeled with the density matrix formalism.
It may be misleading to say that entanglement refers to a particle reacting to the observation of another particle's state with which it is entangled. Instead, the entanglement refers to a quantum correlation that already exists between these particles even before the observation. The observation merely reveals the specific state of the other particle that is correlated with that of the observed particle.
The concept of entanglement comes from the EPR paper and was later developed further by Schroedinger. The term entanglement was coined by him.
The EPR paradox was later formalised in terms of Bell's inequality. Then Alain Aspect (among others) showed that nature violates Bell's inequality. By implication nature allows states to be quantum entangled.
