Short answers, pulled from the story.
Lagrangian field theory is the field-theoretic analogue of Lagrangian mechanics. While Lagrangian mechanics analyzes systems of discrete particles each with a finite number of degrees of freedom, Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
A primary motivation was to provide a rigorous mathematical foundation for quantum field theory, which is formally beset by difficulties that make it unacceptable as a mathematical theory. The classical formulation enables solutions on well-characterized spaces such as Sobolev spaces and allows theorems from geometry, including the Atiyah-Singer index theorem and Chern-Simons theory, to be applied.
Lagrangian density terms divide into mass terms (two fields, no derivatives, which give mass to fields), kinetic terms (two fields with at least one derivative, which make fields dynamical), interaction terms (more than two fields, whose coupling constants set interaction strength), tadpole terms (one field), and constant terms (no fields, which act as a cosmological constant in gravitational theories).
Newtonian gravity, classical electromagnetism, quantum electrodynamics, quantum chromodynamics, general relativity, the Yang-Mills equations, Ginzburg-Landau theory for superconductors, and the Chern-Simons functional can all be expressed as Lagrangian densities. Maxwell's equations, Gauss's law for gravity, and the Einstein field equations each follow directly from varying the appropriate Lagrangian.
The Skyrmion is a topological soliton solution that arises naturally in sigma models, which are a class of Lagrangian field theories describing particles constrained to move on curved manifolds. It serves as a model of the nucleon, the proton or neutron, and is notable for having withstood the test of time as a physical model.
Kaluza-Klein theory unifies electromagnetism and gravity by introducing a fifth dimension and constructing an affine bundle. Early results showing that the seven-sphere could be factored as a product of the four-sphere and the three-sphere generated excitement, but the seven-sphere proved not large enough to contain all of the Standard Model, ending hopes for that approach as a complete unified theory.