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The Dirac equation (iγᵘ∂ᵤψ − mψ = 0) is the relativistic wave equation for spin-½ fermions. It predicts the electron's g-factor and positrons. Solutions are 4-component spinors; spin emerges as a relativistic effect.
You know the Schrödinger equation and how Pauli matrices represent spin-½ as two-component spinors. The problem Dirac faced was that the Schrödinger equation is not Lorentz-invariant: it is first-order in time but second-order in space, treating time and space asymmetrically. The Klein-Gordon equation (∂²ψ/∂t² = ∇²ψ − m²ψ) fixes this by being second-order in both time and space, but it admits negative-energy solutions and a probability density that is not positive-definite — problems that cannot be patched without a fundamentally different approach.
Dirac's insight was to demand an equation that is *first-order in both time and space*, so that the continuity equation automatically gives a positive-definite probability density. To write E = √(p²c² + m²c⁴) as a first-order operator, he needed to "take the square root" of the operator p²c² + m²c⁴ algebraically. He needed matrices α_i and β satisfying anticommutation relations {α_i, α_j} = 2δᵢⱼ and {α_i, β} = 0. Here your knowledge of Pauli matrices becomes essential: these relations cannot be satisfied by 2×2 matrices alone — the minimum size is 4×4. The gamma matrices γᵘ are built from Pauli matrices in block form (the Dirac or Weyl representation being two common choices).
The consequence of needing 4×4 matrices is that the wave function must be a 4-component spinor ψ — this is not an assumption but is forced by the algebra of Lorentz symmetry. The four components split naturally: two correspond to spin-up and spin-down states of the particle, and two to spin-up and spin-down states of the *antiparticle*. Most remarkably, spin-½ emerges automatically from Lorentz symmetry — it is not added by hand as in the non-relativistic Pauli theory. This is one of the deepest results in physics: half-integer spin is an inevitable consequence of combining quantum mechanics with special relativity.
The Dirac equation made two spectacular predictions at the time of its discovery. First, the electron's anomalous magnetic moment: the Schrödinger–Pauli theory assumes g = 2; the Dirac equation *derives* g = 2 (plus small corrections from quantum electrodynamics) from first principles, with no free parameters. Second, the negative-energy solutions — initially troubling — were reinterpreted as antiparticles. Dirac predicted the positron in 1928, before its experimental discovery in 1932. The original "Dirac sea" picture (negative-energy states are all filled, a hole is a positron) has since been superseded by quantum field theory, where positrons are excitations of the electron field; but the core prediction — that every charged fermion has an antiparticle with opposite charge — follows directly from the structure of the equation and has been confirmed for every known fermion.