Perhaps the most beautiful and easy to understand field theory is that of electrodynamics. In the late 1800's, James Clerk Maxwell unified the rather disparate and seemingly unrelated topics of magnetism and electricity into a single theory, codified in the 4 equations below, Maxwell's equations:
\nabla\cdot\vec{B}&=0\\
\nabla\times\vec{E}&=-\frac{1}{c}\frac{\partial\vec{B}}{\partial t}\\
\nabla\times\vec{B}&=\frac{4\pi}{c}\vec{J}+\frac{1}{c}\frac{\partial\vec{E}}{\partial t}\end{align*}\]
The first equations is Gauss's Law, the second precludes the existence of magnetic monopoles, the third is Faraday's Law of induction, and the fourth is the Maxwell-Ampère law with a displacement current. These four partial differential equations can be modified slightly to be used in polarized, dielectric or magnetic materials, but this is essentially the entire theory of electromagnetism. To put these in a covariant framework (for relativistically invariant physics), we can introduce the rank 2 antisymmetric field tensor:
\[F_{\alpha\beta}=\begin{pmatrix} 0 & -E_x & -E_y & -E_z\\ E_x & 0 & B_z & -B_y\\ E_y & -B_z & 0 & B_x\\ E_z & B_y & -B_x & 0\end{pmatrix}\]
Using this and the definition \(J^{\alpha}=(c\rho,\vec{J})\), Maxwell's equations are \partial_{\alpha}F^{\alpha\beta}=\frac{4\pi}{c}J^{\beta}\) and \(\partial_{\gamma}F_{\alpha\beta}+\partial_{\beta}F_{\gamma\alpha}+\partial_{\alpha}F_{\beta\gamma}=0\). In terms of the 4-potential \(A^{\alpha}=(\phi,\vec{A})\), where \(F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\), it simplifies even more: \(\partial^2A^{\alpha}=\frac{4\pi}{c}J^{\alpha}\).
But how can we get this from a Lagrangian? Once again, we must construct some kind of scalar from the fields. From the field tensor, there is only one Lorentz invariant we may make: \(F_{\alpha\beta}F^{\alpha\beta}\). (There is also a pseudo-scalar invariant related to the angle between the electric and magnetic fields, but the Lagrangian must be a true scalar). The only scalar we can form from the current is \(A_{\alpha}J^{\alpha}\), so we choose as our Lagrangian \(\mathcal{L}=-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}+\frac{4\pi}{c}A_{\alpha}J^{\alpha}\), where the constants are chosen to conform to experiment. Invoking Hamilton's Principle by finding a stationary action,
\[\delta A=\delta\int\left[-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}+\frac{4\pi}{c}A_{\alpha}J^{\alpha}\right]\,d^4x=0\]Using the Euler-Lagrange equations exactly return the results above, so that this Lagrangian completely encapsulates electromagnetism. It is straightforward (if cumbersome) to generalize E&M to curved spacetime in the theory of general relativity.