Quantum electrodynamics (QED) is the quantum field theory believed to describe electromagnetic interaction. It is the simplest example of a quantum gauge theory, where the gauge group is abelian, U(1).
In QED, electrically charged particles are coupled by an uncharged, massless vector boson called the photon. The former are described by means of a fermionic spinor field $\psi$, and the latter by a bosonic vector (gauge) field $A$. The classical Lagrangian is postulated to be $$ \mathcal L=\bar\psi(i\not D-m)\psi+\frac{1}{4e^2}F^2 $$ where $D\equiv\partial-iA$ is the so-called gauge covariant derivative and $F\equiv \mathrm dA$ is the so-called field strength tensor. The quantum Lagrangian requires several modifications, such as fixing the gauge and introducing renormalisation constants. Once this is done, one may read off from $\mathcal L$ the Feynman rules of the theory, which are enough to calculate any prediction to an arbitrary order in perturbation theory.
By adding three additional, massive vector bosons (the $Z^0$ and $W^{\pm}$) which couple to the weak hyper-charge ($T_3 - q \ \sin^2 \theta_W$ in which $q$ is electric charge and $T_3$ is the third component of the weak isospin), the theory can be extended to cover the weak nuclear force as well.
