I’ll give you a brief over-view of the Dirac equation:
\sum_{\mu}\left(i\hbar\gamma^{\mu}\partial_{\mu}-mc)\psi =0
\hbar is Plank's constant over two pi, \partial_{\mu} a partial derivative (a derivative of a function of more than one variable with respect to a single variable) with respect to the subscripted variable (\mu =t, x, y, z), m the mass, c the speed of light and i=\sqrt{-1}. \psi is a Dirac/Weyl spinor that is a more complicated variety of wave-function than that found in simple non-relativistic quantum mechanics, it's a bit like a vector but they transform under SU(2) rather than SO(N), meaning they're a vector in a complex (involving roots of negatives) vector space. \gamma^{\mu} is the mu-th component of a set of matrices that transform as a vector. They form a Clifford algebra with the following anti-commutation relation in Minkowski space-time (the space-time of special relativity with metric diag(1 -1 -1 -1))
\left\{\gamma^{\mu}, \gamma^{\nu}\right\} =2\eta^{\mu\nu}\times\mathbf{1}_{n\times n}
\eta^{\mu\nu} is the symmetric (same under interchange of the indices mu and nu) Minkowskian metric tensor, the braces with a comma denotes the commutator \{a, b\}=ab+ba and \mathbf{1}_{n\times n} is the identity operator with dimension n equal to the representation of the algebra.
You'll find a discussion of the Dirac equation and its results in any text that covers relativistic quantum mechanics and/or field theory. Its use in quantum field theory is of great importance, as it accurately describes the dynamics of fermions that are both free of interactions and involved in some form of interaction (with the appropriate coupling to the interaction and other terms that come from those in the Lagrangian that are gauge invariant under the symmetry group of the field theory).
To get the best results from the Dirac theory we need to apply a method of quantisation to the Dirac field \psi (particle can only take on set, discrete amounts of energy). If we don't use quantum field theory and minimally couple the Dirac equation (whack in a gauge connection term that accounts for the interaction's coupling to the field) the Dirac equation can only, at best, account for first order interactions, meaning it can only make calculations from the simplest Feynman diagrams we can draw (known as tree-level, which are absent of virtual particle interactions/bubbles).
The full equation for an electron interacting with an electromagnetic field (Quantum Electrodynamics, QED) is
\mathcal{L}=\sum_{\mu,\nu}\left[\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi-\tfrac{1}{4}F^{\mu\nu}F_{\mu\nu}-e\bar{\psi}\gamma^{\mu}A_{\mu}\psi\right]
Here the Dirac field with the bar over it represents the anti-particle form of the fermion field \psi, e is the charge on the electron, F is the field-strength tensor of classical electrodynamics (=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}) and A is the electromagnetic potential (satisfying \mathbf{E}=\nabla A_0, \mathbf{B}=\nabla\times\mathbf{A}). This quantity is known as a Lagrangian density \cal{L} or simply Lagrangian, which by applying the principle of least action, also known as Hamilton's principle, will give one the equations of motion for the electron/photon. The QED Lagrangian accurately describes all electrodynamic interactions between fermions.
There aren't really any complicated equations from special relativity. The most important equation from general relativity is
G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}
which are a set of 10 second order differential equations in the metric tensor (or connection in Einstein-Cartan theory) known as the Einstein field equations (in my opinion they should really be called the Einstein-Hilbert equations, but nevermind).
