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Generating functionals encode all correlation functions of a quantum field theory in a single object. Z[J] generates all Green's functions, W[J] = -i ln Z[J] generates connected Green's functions, and the effective action Gamma[phi_cl] generates one-particle-irreducible (1PI) vertices. These functionals provide a compact and powerful framework for deriving Ward identities, the effective potential, and non-perturbative results.
The generating functional Z[J] = integral D[phi] ei(S[phi] + integral J phi d4x) is the master object of quantum field theory. Every correlation function -- every Green's function, every scattering amplitude -- is obtained by taking functional derivatives of Z with respect to the external source J(x). The n-point Green's function is Gn(x1, ..., xn) = (-i)n (deltan Z / delta J(x1)...delta J(xn))|_{J=0} / Z[0]. Having all information in a single functional allows you to derive general relations (like Ward identities) that constrain all correlation functions simultaneously.
The connected generating functional W[J] = -i ln Z[J] generates only the connected Green's functions -- those where all external points are linked by a chain of propagators. Disconnected Green's functions are products of lower-point connected functions and contain no new information. W[J] is the analog of the cumulant generating function in statistics: the logarithm strips off the factorizable part. This is physically relevant because the S-matrix depends only on connected diagrams (the linked cluster theorem).
The most powerful object is the effective action Gamma[phi_cl], defined as the Legendre transform of W[J]: Gamma[phi_cl] = W[J] - integral J phi_cl d4x, where phi_cl(x) = delta W/delta J(x) is the classical field (the vacuum expectation value of the quantum field in the presence of the source). The functional derivatives of Gamma with respect to phi_cl give the one-particle-irreducible (1PI) vertex functions -- the building blocks from which all Green's functions are constructed. The remarkable property of Gamma is that if you knew it exactly, you could compute the full quantum theory using only tree-level Feynman rules with Gamma as the action.
The effective potential V_eff(phi_cl) is the effective action evaluated for constant field configurations: Gamma = -integral V_eff d4x (for spatially uniform fields). It gives the full quantum-corrected potential energy density, including all loop effects. The minima of V_eff determine the true quantum vacuum, which may differ from the classical vacuum. This is how radiative corrections can trigger spontaneous symmetry breaking (Coleman-Weinberg mechanism) or modify the Higgs potential. The effective potential is one of the most direct applications of functional methods to physical questions about the vacuum structure of field theories.