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The S-matrix (scattering matrix) maps initial states of incoming particles to final states of outgoing particles. Its matrix elements encode all observable scattering information. The S-matrix is decomposed as S = 1 + iT, where T contains the non-trivial scattering amplitude M related to physical cross sections and decay rates.
The S-matrix is the central object connecting quantum field theory to experiment. In a scattering experiment, you prepare an initial state |i> of incoming particles with definite momenta long before the interaction, and you measure the final state |f> of outgoing particles long after. The S-matrix element <f|S|i> gives the probability amplitude for this transition, and the probability is |<f|S|i>|^2. Every measurement in particle physics -- every cross section, branching ratio, and decay rate -- is extracted from S-matrix elements.
The S-matrix is decomposed as S = 1 + iT, where the identity represents the trivial case of no interaction (particles pass through without scattering). The T-matrix encodes the non-trivial scattering. For a specific process, the matrix element is <f|iT|i> = i(2pi)4 delta^4(p_i - p_f) M_{fi}, where the delta function enforces total energy-momentum conservation and M is the invariant amplitude (or Feynman amplitude). The Feynman diagram expansion computes M order by order in the coupling constant: each diagram at a given order contributes a term to M.
Two fundamental properties of the S-matrix constrain all of physics. Unitarity (S-dagger S = 1) is the statement that total probability is conserved: the probabilities of all possible final states must sum to 1. This leads to the optical theorem, which relates the imaginary part of the forward scattering amplitude to the total cross section, and to cutting rules (Cutkosky rules) that relate loop diagrams to products of tree-level diagrams. Lorentz invariance requires that S-matrix elements are the same in all inertial frames, which constrains the form of the amplitude M.
The formal connection between S-matrix elements and the field-theoretic correlation functions is provided by the LSZ reduction formula. It shows that S-matrix elements are obtained from time-ordered Green's functions by going on-shell (setting the external momenta to satisfy the mass-shell condition p2 = m2) and amputating external propagators. This justifies the Feynman diagram approach: you compute the amputated, connected Green's function using Feynman rules, evaluate it with on-shell external momenta, and the result is the scattering amplitude M. The LSZ formula also introduces wave function renormalization factors that account for the difference between the bare fields in the Lagrangian and the physical particle states.