Intuitive

The Dirac equation describes how the state of a relativistic (= fast moving) quantum system with half-integer spin changes in time.

The analogous equation for systems without spin is the Klein-Gordon equation.

If the system only moves slowly, the Dirac equation becomes the Pauli equation.

Concrete

• A nice discussion can be found in chapter 4 of Klauber's Student Friendly QFT book
• For an elementary derivation and a discussion of what the solution of the Dirac equation mean, see Physics from Symmetry by Schwichtenberg
• Alternative, two possible derivations can be found at page 100ff in Relativistic Quantum Mechanics by Paul Strange
• A great discussion can be found in chapter 10 of the quantum field theory book by Matthew Schwartz.
• Another great discussion, especially of question how many degrees of freedom a Dirac spinor has can be found in http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf

Gamma Gymnastics:

There are many important rules for the $\gamma$ matrices that appear in the Dirac equation. These rules are important for many practical calculations.

• For a nice description, see section 7.4.3 "Diracology" in the book The Conceptual Framework of Quantum Field Theory by Duncan

Graphical Summary

The diagram below shows the Dirac equation and its Lagrangian in various forms. For a more detailed explanation see Fun with Symmetry.