Short answers, pulled from the story.
Claude Itykson and Jean-Bernard Zuber published a text in 1980 that defined the modern approach to Lagrangian field theory. This formalism serves as the field-theoretic analogue of Lagrangian mechanics used for discrete particles.
The independent variable in field theory is replaced by an event in spacetime or a point on a Riemannian manifold. Field theories apply to continua and fields with an infinite number of degrees of freedom unlike discrete particle systems which possess a finite number of degrees of freedom.
The Lagrange density for general relativity includes the cosmological constant and curvature scalar within the integral known as the Einstein Hilbert action involving Riemann tensor contractions. Substituting this Lagrangian into Euler Lagrange equations yields Einstein field equations when taking the metric tensor as the field.
Terms containing the product of two fields and no derivatives are known as mass terms because they give mass to the fields themselves. Other terms with at least one derivative are known as kinetic terms that make fields dynamical while most theories restrict them to at most two derivatives.
The A field can be understood as the affine connection on a U(1)-fiber bundle over spacetime. This formulation allows writing the electromagnetic action without an additional integration measure using differential forms that have coordinate differentials built directly into their structure.